unstable surface waves in running waterpeople.math.gatech.edu/~zlin/publication/hurlin.pdf · 2007....

UNSTABLE SURFACE WAVES IN RUNNING WATER

VERA MIKYOUNG HUR AND ZHIWU LIN

Abstract. We consider the stability of periodic gravity free-surface waterwaves traveling downstream at a constant speed over a shear flow of finitedepth. In case the free surface is flat, a sharp criterion of linear instabilityis established for a general class of shear flows with inflection points and themaximal unstable wave number is found. Comparison to the rigid-wall settingtestifies that free surface has a destabilizing effect. For a class of unstable shearflows, the bifurcation of nontrivial periodic traveling waves of small-amplitudeis demonstrated at any wave number. We show the linear instability of smallnontrivial waves bifurcated at an unstable wave number of the backgroundshear flow. The proof uses a new formulation of the linearized water-waveproblem and a perturbation argument. An example of the background shearflow of unstable small-amplitude periodic traveling waves is constructed foran arbitrary vorticity strength and for an arbitrary depth, illustrating thatvorticity has a subtle influence on the stability of water waves.

1. Introduction

The water-wave problem in its simplest form concerns two-dimensional motionof an incompressible inviscid liquid with a free surface, acted on only by gravity.Suppose, for definiteness, that in the (x, y)-Cartesian coordinates gravity acts inthe negative y-direction and that the liquid at time t occupies the region boundedfrom above by the free surface y = η(t; x) and from below by the flat bottom y = 0.In the fluid region (x, y) : 0 < y < η(t;x), the velocity field (u(t; x, y), v(t; x, y))satisfies the incompressibility condition

(1.1) ∂xu + ∂yv = 0

and the Euler equation

(1.2)

∂tu + u∂xu + v∂yu = −∂xP

∂tv + u∂xv + v∂yv = −∂yP − g,

where P (t;x, y) is the pressure and g > 0 denotes the gravitational constant ofacceleration. The flow is allowed to have rotational motions and characterized bythe vorticity ω = vx − uy. The kinematic and dynamic boundary conditions at thefree surface y = η(t;x)(1.3) v = ∂tη + u∂xη and P = Patm

express, respectively, that the boundary moves with the velocity of the fluid par-ticles at the boundary and that the pressure at the surface equals the constantatmospheric pressure Patm. The impermeability condition at the flat bottom statesthat

(1.4) v = 0 at y = 0.1

2 VERA MIKYOUNG HUR AND ZHIWU LIN

It is a matter of common experience that waves which may be observed on thesurface of the sea or on the river are approximately periodic and propagating ofpermanent form at a constant speed. In case ω ≡ 0, namely in the irrotationalsetting, waves of the kind are referred to as Stokes waves, whose mathematicaltreatment was initiated by formal but far-reaching considerations of Stokes [43]himself. The existence theory of Stokes waves dates back to the construction dueto Levi-Civita [30] and Nekrasov [40] in the infinite-depth case and due to Struik[44] in the finite-depth case of small-amplitude waves, and it includes the globaltheory due to Krasovskii [29] and Keady and Norbury [28]. Stokes waves of greatestheight exist [47], [37] and are shown to have stagnation at wave crests [4]. Whilethe irrotationality assumption may serve as an approximation under certain circum-stances and has been used in a majority of the existing research, surface water-wavestypically carry vorticity, e.g. shear currents on a shallow channel and wind-driftboundary layers. Moreover, the governing equations for water waves allow for ro-tational steady motions. Gersner [21] early in 1802 found an explicit formula for afamily of periodic traveling waves on deep water with a particular nonzero vortic-ity. An extensive existence theory of periodic traveling water waves with vorticityappeared in the construction due to Dubreil-Jacotin [20] of small-amplitude waves.Recently, for a general class of vorticity distributions, Constantin and Strauss [14]in the finite-depth case and Hur [26] in the infinite-depth case accomplished thebifurcation analysis for periodic traveling waves of large amplitude.

Waves of Stokes’ kind is one of the few exact solutions of the free-surface water-wave problem, and as such it is important to understand the stability of thesesolutions. The present purpose is to investigate the linear instability of periodicgravity water-waves with vorticity.

The stability of water waves in case of zero vorticity has been under researchmuch by means of numerical computations and formal analysis, especially in theworks of Longuet-Higgins and his coworkers. Numerical studies of Stokes wavesunder perturbations of the same period, namely the superharmonic perturbations,indicate that [39], [45] instability sets in only when the wave amplitude is largeenough to link with wave breaking [18], and thus small-amplitude Stokes wavesare found to be linearly stable under the same-period perturbations. MacKay andSaffman [36] also considered linear stability of small-amplitude Stokes waves bymeans of general results of the Hamiltonian system. The Hamiltonian formula-tion in terms of the velocity potential, however, does not avail in the presence ofeffects of vorticity. The analysis of Benjamin and Feir [9] showed that there is a“sideband” instability for small Stokes waves, meaning that the perturbation hasa different period than the steady wave. The Benjamin-Feir instability was mademathematically rigorous by Bridges and Mielke [10].

Analytical works on the stability of water waves with vorticity, on the otherhand, are quite sparse. A recent contribution due to Constantin and Strauss [15]concerns two different kinds of formal stability of periodic traveling water-waveswith vorticity under perturbations of the same period. First, in case when thevorticity decreases with depth an energy-Casimir functional H is constructed asa temporal invariant of the nonlinear water-wave problem, whose first variationgives the exact equations for steady waves. Its second variation is positive and thewater-wave system is H-formally stable under some special perturbations. Theirsecond approach uses another functional J , which is essentially the dual of H in

UNSTABLE SURFACE WAVES IN RUNNING WATER 3

the transformed variables but not an invariant. Its first variation gives the exactequations for steady waves in the transformed variables, which served as a basis in[14] of the existence theory for traveling waves. The J -formal stability then meansthe positivity of its second variation. The “exchange of stability” theorem due toCrandall and Rabinowitz [16] applies to conclude that the J -formal stability of thetrivial solutions switches exactly at the bifurcation point and that steady wavesalong the curve of local bifurcation are J -formally stable provided that both thedepth and the vorticity strengh are sufficiently small.

The main results. As a preliminary step toward the stability and instabilityof nontrivial periodic waves, we examine the linear stability and instability of flat-surface shear flows. The stability of shear flows in the rigid-wall setting is a classicalproblem [19], whose theories date back to the necessary condition due to Rayleigh[42]. Recently, Lin [32] obtained instability criteria for several classes of shear flowsin a channel with rigid walls, and our results generalize these to the free-surfacesetting. More specifically, our conclusions include: (1) The linear stability of shearflows with no inflection points (Theorem 6.4), which generalizes Rayleigh’s criterionin the rigid-wall setting [42] to the free-surface setting; (2) A sharp criterion of linearinstability for a class of shear flows with one inflection value (Theorem 4.2); and(3) A sufficient condition of linear instability for a class of shear flows with multipleinflection values (Theorem 6.1) including monotone flows. Our result testifies thatfree surface has a destabilizing effect compared to rigid walls.

Our next step is to understand the local bifurcation of small-amplitude periodictraveling waves in the physical space. While our setting is similar to that in [15] inthat it hinges on the existence results of periodic waves in [14] via local bifurcation,the choice of bifurcation parameter and the dependence of other parameters on thebifurcation parameter and free parameters in the description of the backgroundshear flow are different. In our setting, it is natural to consider that the shearprofile and the channel depth are given and that the speed of wave propagationis chosen to ensure local bifurcation. The relative flux and the vorticity-streamfunction relation are then computed. In contrast, in the bifurcation analysis [14] inthe transformed variables, the wave speed is given arbitrary and the relative fluxand the vorticity-stream function relation are held fixed. In turn, the shear profileand the channel depth vary along the bifurcation curve. Lemma 2.3 establishesthe equivalence between the bifurcation equation (2.7) in the transformed variablesand the Rayleigh system (2.9)–(2.10) to obtain the bifurcation results for a largeclass of shear flows. In addition, our result helps to clarify that the nature of thelocal bifurcation of periodic traveling water-waves does not involve the exchange ofstability of trivial solutions (Remark 4.14).

Our third step is to show under some technical assumptions that the linearinstability of the background shear flow persists along the local curve of bifurcationof small-amplitude periodic traveling waves (Theorem 5.1). An example of such anunstable shear flow is

U(y) = a sin b(y − h/2) for y ∈ [0, h],

where h, b > 0 satisfy hb 6 π and a > 0 is arbitrary (Remark 5.2). In particular, bychoosing a and h to be arbitrarily small, we can construct linearly unstable smallperiodic traveling water-waves with an arbitrarily small vorticity strength and anarbitrarily small channel depth. This indicates that the formal stability of the


second kind in [15] is quite different from the linear stability of the physical waterwave problem. Our example also shows that adding an arbitrarily small vorticity tothe water-wave system may affect the superharmonic stability of small-amplitudeperiodic irrotational waves in a water of arbitrary depth. Thus, it is important totake into account of the effects of vorticity in the study of the stability of waterwaves.

Temporal invariants of the linearized water-wave problem are derived and theirimplications for the stability of the water-wave system are discussed (Section 3.3).In case when the vorticity-stream function relation is monotone, the energy func-tional ∂2H in [15] is indeed an invariant of the linearized water-wave problem.Other invariants are derived, and one may study the stability of the water-waveproblem via the energy method. However, even with these additional invariants asconstraints, the quadratic form ∂2H is in general indefinite, indicating that thata steady (gravity) water-wave may be an energy saddle. A successful proof of thestability for the water-wave problem therefore would require to exhaust the fullequations instead of using a few of its invariants.

Ideas of the proofs. Our approach in the proof of the linear instability of free-surface shear flows uses the Rayleigh system (4.1)–(4.2), which is related to thatin the rigid-wall setting [32]. The main difference from [32] lies in the complicatedboundary condition (4.2) on the free surface, which renders the analysis more in-volved. The instability property depends on the wave number, which is consideredas a parameter. As in the rigid-wall setting [32], a key to a successful instabilityanalysis is to locate the neutral limiting modes, which are neutrally stable solutionof the Rayleigh system and contiguous to unstable modes. For certain classes offlows, neutral limiting modes in the free-surface setting are characterized by the in-flection values. This together with the local bifurcation of unstable solutions fromeach neutral limiting wave number gives a complete knowledge on the instabilityat all wave numbers.

The instability analysis of small-amplitude nontrivial waves taken here is basedon a new formulation which directly linearizes the Euler equation and the kinematicand dynamic boundary conditions on the free surface around a periodic travelingwave. Its growing-mode problem then is written as an operator equation for thestream function perturbation restricted on the steady free-surface. The mapping bythe action-angle variables is employed to prove the continuity of the operator withrespect to the amplitude parameter. In addition, in the action-angle variables, theequation of the particle trajectory becomes simple. The persistence of instabilityalong the local curve of bifurcation is established by means of Steinberg’s eigen-value perturbation theorem [41] for operator equations. In addition, growing-modesolutions of the operator equation acquire regularity.

This paper is organized as follows. Section 2 is the discussion on the localbifurcation of periodic traveling water-waves when a background shear flow in thephysical space is given. Section 3 includes the formulation of the linearized periodicwater-wave problem and the derivation of its invarints. Section 4 is devoted to thelinear instability of shear flows with one inflection value, and subsequently, Section5 is to the linear instability of small-amplitude periodic waves over an unstableshear flow. Section 6 revisits the linear instability of shear flows in a more generalclass.


2. Existence of small-amplitude periodic traveling water-waves

We consider a traveling-wave solution of (1.1)–(1.4), that is, a solution for whichthe velocity field, the wave profile and the pressure have space-time dependence(x−ct, y), where c > 0 is the speed of wave propagation. With respect to a frame ofreference moving with the speed c, the wave profile appears to be stationary and theflow is steady. The traveling-wave problem for (1.1)–(1.4) is further supplementedwith the periodicity condition that the velocity field, the wave profile and thepressure are 2π/α-periodic in the x-variable, where α > 0 is the wave number.

It is traditional in the traveling-wave problem to define the relative stream func-tion ψ(x, y):

(2.1) ψx = −v, ψy = u− c

and ψ(0, η(0)) = 0. This reduces the traveling-wave problem for (1.1)–(1.4) to astationary elliptic boundary value problem [14, Section 2]:

For a real parameter B and a function γ ∈ C1+β([0, |p0|]), β ∈ (0, 1), find η(x)and ψ(x, y) which are 2π/α-periodic in the x-variable, ψy(x, y) < 0 in (x, y) : 0 <y < η(x)1, and

−∆ψ = γ(ψ) in 0 < y < η(x),(2.2a)

ψ = 0 on y = η(x),(2.2b)

|∇ψ|2+2gy = B on y = η(x),(2.2c)

ψ = −p0 on y = 0,(2.2d)

where

(2.3) p0 =∫ η(x)

0

ψy(x, y)dy

is the relative total flux2.The vorticity function γ gives the vorticity-stream function relation, that is,

ω = γ(ψ). The assumption of no stagnation, i.e. ψy(x, y) < 0 in the fluid region(x, y) : 0 < y < η(x), guarantees that such a function is well-defined globally;See [14]. Furthermore, under this physically motivated stipulation, interchangingthe roles of the y-coordinates and ψ offers an alternative formulation to (2.2) ina fixed strip, which serves as the basis of the existence theories in [20], [14], [26].The nonlinear boundary condition (2.2c) at the free surface y = η(x) expressesBernoulli’s law. The steady hydrostatic pressure in the fluid region is given by

(2.4) P (x, y) = B − 12 |∇ψ(x, y)|2 − gy −

∫ ψ(x,y)

0

γ(−p)dp.

In this setting, α and B are considered as parameters whose values form part ofthe solution. The wave number α in the existence theory is independent of otherphysical parameters and hence is held fixed, while in the stability analysis in Section4 it serves as parameter. The Bernoulli constant B measures the total mechanicalenergy of the flow and varies along a solution branch.

1In other words, there is no stagnation in the fluid region. Field observations [31] as well aslaboratory experiments [46] indicate that for wave patterns which are not near the spilling orbreaking state, the speed of wave propagation is in general considerably larger than the horizontalvelocity of any water particle.

2p0 < 0 is independent of x.


2.1. The local bifurcation theorem in [14]. This subsection contains a sum-mary of the existence result in [14] via the local bifurcation theorem of small-amplitude travelling-wave solutions to (2.2) provided that the total flux p0 and thevorticity-stream function relation γ are given.

A preliminary result for local bifurcation is to find a curve of trivial solutions,which correspond to horizontal shear flows under a flat surface. As in [14, Section3.1], let

Γ(p) =∫ p

0

γ(−p′)dp′, Γmin = min[p0,0]

Γ(p) 6 0.

Lemma 2.1 ([14], Lemma 3.2). Given p0 < 0 and γ ∈ C1+β([0, |p0|]), β ∈ (0, 1),for each µ ∈ (−2Γmin,∞) the system (2.2) has a solution

y(p) =∫ p

p0

dp′√µ + 2Γ(p′)

,

which corresponds to a parallel shear flow in the horizontal direction

(2.5) u(t;x, y) = U(y;µ) = c−√

µ + 2Γ(p(y))

and v(t;x, y) ≡ 0 in the channel (x, y) : 0 < y < h(µ), where

h(µ) =∫ 0

p0

dp√µ + 2Γ(p)

.

The hydrostatic pressure is P (y) = −gy for y ∈ [0, h(µ)]. Here, p(y) is the inverseof y = y(p) and determines the stream function ψ(y; µ) = −p(y;µ); c > 0 isarbitrary.

In the statement of Theorem 2.2 below, instead of B the squared (relative)upstream flow speed µ = (U(h)−c)2 of a trivial shear flow (2.5) serves as a bifurca-tion parameter. For each µ ∈ (−2Γmin,∞) the Bernoulli constant B is determineduniquely in terms of µ as

(2.6) B = µ + 2g

∫ 0

p0

dp√µ + 2Γ(p)

.

The following theorem [14, Theorem 3.1] states the existence result of a one-parameter curve of small-amplitude periodic water-waves for a general class of vor-ticities and their properties, in a form convenient for our purposes.

Theorem 2.2 (Existence of small-amplitude periodic water-waves). Let the speedof wave propagation c > 0, the flux p0 < 0, the vorticity function γ ∈ C1+β([0, |p0|]),β ∈ (0, 1), and the wave number α > 0 be given such that the system

(2.7)

(a3(µ)Mp)p = α2a(µ)M for p ∈ (p0, 0)µ3/2Mp(0) = gM(0)M(p0) = 0

admits a nontrivial solution for some µ0 ∈ (−2Γmin,∞), where a(µ) = a(µ; p) =√µ + 2Γ(p).Then, for ε > 0 sufficiently small there exists a one-parameter curve of steady

solution-pair µε of (2.6) and (ηε(x), ψε(x, y)) of (2.2) such that ηε(x) and ψε(x, y)are 2π/α-periodic in the x-variable, of C3+β class, where β ∈ (0, 1), and ψεy(x, y) <0 throughout the fluid region.


At ε = 0 the solution corresponds to a trivial shear flow under a flat surface:(i0) The flat surface is given by η0(x) ≡ h(µ0) =: h0 and the velocity field is

(ψ0y(x, y),−ψ0x(x, y)) = (U(y)− c, 0),

where U(y) is determined in (2.5);(ii0) The pressure is given by the hydrostatic law P0(x, y) = −gy for y ∈ [0, h0].At each ε > 0 the corresponding nontrivial solution enjoys the following proper-

ties:(iε) The bifurcation parameter has the asymptotic expansion

µε = µ0 + O(ε) as ε → 0

and the wave profile is given by

ηε(x) = hε + α−1δγε cosαx + O(ε2) as ε → 0,

where hε = h(µε) is given in Lemma 2.1 and δγ depends only on γ and p0;The mean height satisfies

hε = h0 + O(ε) as ε → 0;

Furthermore, the wave profile is of mean-zero; That is,

(2.8)∫ 2π/α

0

(ηε(x)− hε)dx = 0;

(iiε) The velocity field (ψεy(x, y),−ψεx(x, y)) in the steady fluid region (x, y) :0 < x < 2π/α , 0 < y < ηε(x) is given by

ψεx(x, y) = εψ∗x(y) sin αx + O(ε2),

ψεy(x, y) = U(y)− c + εψ∗y(y) cos αx + O(ε2)

as ε → 0, where ψ∗x and ψ∗y are determined from the linear theory;(iiiε) The hydrostatic pressure has the asymptotic expansion

Pε(x, y) = −gy + O(ε) as ε → 0.

The condition that the system (2.7) admits a nontrivial solution for some µ0 ∈(−2Γmin,∞) is necessary and sufficient for local bifurcation [14, Section 3]. A suffi-cient condition ([14]) for the solvability of (2.7) and therefore the local bifurcationis ∫ 0

p0

(α2(p− p0)2(2Γ(p)− 2Γmin)1/2 + (2Γ(p)− 2Γmin)3/2

)dp < gp2

0,

which is satisfied when p0 is sufficiently small.

2.2. The bifurcation condition for shear flows. Our instability analyses inSection 4 and Section 5 are carried out in the physical space, where a shear-flowprofile and the depth are held fixed. The bifurcation analysis in the proof of The-orem 2.2, on the other hand, is carried out in the space of transformed variables,where the relative flux p0 and the vorticity-stream function relation γ are held fixed,and the relative flow speed U(h)− c and the channel height h vary along the curveof local bifurcation. In this subsection, we undertake the study of the local bifurca-tion in the physical space, which is relevant to the future instability analyses. Thenatural choice for parameters is the speed of wave propagation c > max U and thewave number k.


Our first task is to relate the bifurcation equation (2.7) in transformed variableswith the Rayleigh system in the physical variables.

Lemma 2.3. For the shear flow U(y) for y ∈ [0, h] defined via Lemma 2.1, thebifurcation equation (2.7) is equivalent to that the following Rayleigh equation

(2.9) (U − c)(φ′′ − k2)φ− U ′′φ = 0 for y ∈ (0, h)

with the boundary conditions

(2.10) φ′(h) =(

g

(U(h)− c)2+

U ′(h)U(h)− c

)φ(h) and φ(0) = 0,

where (c, k) = (c, α), c > max U , and φ(y) = (c − U(y))M(p(y)). Here and in thesequel, the prime denotes the differentiation in the y-variable.

Proof. Notice that c > maxU . Indeed,

a(µ; p) =√

µ + 2Γ(p) = −(U(y(p))− c) > 0.

Since ∂p∂y = −ψ′(y) = −(U(y) − c), it follows that ∂p = ∂y

∂y∂p = − 1

U−c∂y. LetM(p(y)) = Φ(y), then (2.7) is written as

((U − c)2Φ′)′ − α2(U − c)2Φ = 0 for y ∈ (0, h),

(U − c)2Φ′(h) = gΦ(h) and Φ(0) = 0.

Let φ(y) = (c− U(y))Φ(y), and the above system becomes (2.9)–(2.10). ¤

Remark 2.4. We illustrate how to construct downstream-traveling periodic waves ofsmall-amplitude bifurcating from a fixed background shear-flow U(y) for y ∈ [0, h].First, one finds the parameter values (c, k) = (c, α) with c > max U and α > 0such that the Rayleigh system (2.9)– (2.10) admits a nontrivial solution. The wavespeed then determines the bifurcation parameter as µ0 = (U(h)− c)2. the flux andthe vorticity function determined as

(2.11) p0 =∫ h

0

(U(y)− c)dy and γ(p) = U ′(y(p)),

respectively. In view of Lemma 2.3 the bifurcation equation (2.7) with µ0, p0 andγ are as above has a nontrivial solution. Moreover, each U(h) − c for c > max Ucorresponds to a trivial solution in Lemma 2.1. Indeed, µ and c has a one-to-onecorrespondence µ = (U(h)− c)2; The (relative) stream function defined as

ψ(y) = −∫ h

y

(U(h)− c)dy

is monotone and its inverse y = y(−ψ) is well-defined. Then, Theorem 2.2 appliesin the setting above to obtain the local curve of bifurcation of periodic waves.

The lemma below obtains for a large class of shear flows the local bifurcation byshowing that the Rayleigh system (2.9)–(2.10) has a nontrivial solution.

Lemma 2.5. If

(2.12) U ∈ C2([0, h]), U ′′(h) < 0 and U(h) > U(y) for y 6= h,

then for any wave number k > 0 there exists c(k) > U(h) = max U such that thesystem (2.9)–(2.10) has a nontrivial solution φ with φ > 0 in (0, h].


Proof. For c ∈ (U(h),∞) and k > 0 let φc be the solution of (2.9), or equivalently,

(2.13) ((U − c)φ′c − U ′φc)′ − k2(U − c)φc = 0 for y ∈ (0, h)

with φc(0) = 0 and φ′c(0) = 1. Integrating the above equation on the interval [0, h]yields that

(U(h)− c)φ′c(h)− (U(0)− c)− φc(h)U ′(h)− k2

∫ h

0

(U − c)φcdy = 0.

Note that the bifurcation condition (2.10) is fulfilled if and only if the function

(2.14) f(c) = c− U(0) + k2

∫ h

0

(c− U)φcdy − g

c− U(h)φc(h)

has a zero at some c(k) > U(h). It is easy to see that f is a continuous function ofc whenever c > U(h).

We claim that φc(y) > 0 for y ∈ (0, h]. Suppose, on the contrary, that φc(y0) = 0for some y0 ∈ (0, h]. Note that (2.9) is written as a Sturm-Liouville equation as

(2.15) φ′′c − k2φc − U ′′

U − cφc = 0.

Since

(c− U)′′ +U ′′

c− U(c− U) = 0 for y ∈ (0, h),

Sturm’s comparison theorem applies to assert that the function c−U(y) must havea zero on the interval (0, y0). A contradiction then proves the claim. Our goal is toshow that f(c) > 0 for c large enough and f(c) < 0 as c → U(h)+.

First, as c → ∞ the sequence of a solution φc of (2.9), or equivalently (2.15)converges in C2, that is φc → φ∞. By continuity, φ∞ satisfies the boundary valueproblem

φ′′∞ − k2φ∞ = 0 for y ∈ (0, h)

with φ∞(0) = 0 and φ′∞(0) = 1. Therefore, φ∞ is bounded, continuous, andpositive on (0, h]. By the definition (2.14) then it follows that f(c) →∞ as c →∞.

Next is to examine f(c) as c → U(h)+. Let us denote ε = c− U(h) > 0 be thesmall parameter. We claim that φc(h) > C1 > 0 for ε > 0 sufficiently small, whereC1 > 0 is independent of ε. To see this, it is convenient to write (2.13) as

((c− U)φ′c − (c− U)′φc)′ = k2(c− U)φc > 0 for y ∈ (0, h),

whence

(c− U(y))φ′c(y)− (c− U(y))′φc(y) > c− U(0) > 0 for y ∈ (0, h).

Integrating the above yields that

φc(h) > (c− U(h))∫ h

0

c− U(0)(c− U(y))2

dy.

This uses that (c−U)φ′c− (c−U)′φc = (c−U)2(

φc

c−U

)′. Our assumption on U(y)

asserts that 0 6 U(h) − U(y) 6 β(h − y) for y ∈ [h − δ, h] for δ > 0 small, where


β > 0 is a constant. Thus,

φc(h) > ε(c− U(0))∫ h

h−δ

1(ε + β(h− y))2

dy

= (c− U(0))∫ (h−δ)/ε

0

1(1 + βx)2

dx > C1 > 0,

where C1 > 0 is independent of ε > 0. This proves the claim. The treatment of thesecond term in the definition of (2.14) is divided into two cases.

First, in case max φc(y) = φc(h) it follows that

f(c) 6 c− U(0) + C1

(k2

∫ h

0

(c− U)dy − g

c− U(h)

)< 0

provided that c− U(h) = ε > 0 is small enough.Next, in case max φc = φc(yc), where yc ∈ (0, h), since φ′′c (yc) 6 0 and φc(yc) > 0

by (2.15) it follows that U ′′(yc) > 0. On the other hand, U ′′(h) 6 0 and henceyc ∈ [0, h − δ] for some δ > 0. Note that c − U(y) is bounded away from zero onthe interval y ∈ [0, h − δ]. Since the coefficients of (2.15) is uniformly boundedfor c on y ∈ [0, h − δ], the solution φc is bounded on y ∈ [0, h − δ]. In particular,0 < φc(yc) 6 C2 independently for c. Therefore,

f(c) 6 c− U(0) + k2hC2 max[0,h]

(c− U(y))− gC1

c− U(h)< 0

for ε = c − U(h) is small enough, where C1, C2 > 0 are independent of ε. Thiscompletes the proof. ¤

Lemma 2.3 and Remark 2.4 ensure the local bifurcation for a shear flow satisfying(2.12) at any wave number k > 0.

Theorem 2.6. If U ∈ C2([0, h]), U ′′(h) < 0 and U(h) > U(y) for y 6= h thenfor an arbitrary wavelength 2π/k, where k > 0, there exist small-amplitude periodicwaves bifurcating in the sense as in Theorem 2.2 from the flat-surface shear flowU(y), where c(k) > max U .

In the irrotational setting, i.e. U ≡ 0, the parameter values c and k for whichthe Rayleigh system (2.9)– (2.10) is solvable give the dispersion relation ([17], forinstance)

c2 =g tanh(kh)

k.

In case with a nonzero background shear flow, on the other hand, such an explicitalgebraic relation is no longer available. Still, the Rayleigh system (2.9)–(2.10) maybe considered to give a generalized dispersion relation. Moreover, the followingquantitative information about c(k) may be derived.

Lemma 2.7. Given a shear flow U(y) in [0, h], let k and c(k) > max U be suchthat (2.9)–(2.10) has a nontrivial solution. Then,

(a) c(k) is bounded for k > 0;(b) If k1 6= k2 then c(k1) 6= c(k2);(c) In the long wave limit k → 0+, the limit of the wave speed c(0) satisfies

Burns condition [12] ∫ h

0

dy

(U − c(0))2=

1g.


Proof. (a) The proof of Lemma 2.5 implies that for each A > 0 there exists CA suchthat f(c) defined in (2.14) is positive when c > CA and k < A. In interpretation,c(k) 6 CA for k < A. Thus, it suffices to show that f(c) > 0 when c and k arelarge enough. Indeed, let c > max U be large enough so that

∣∣∣ U ′′c−U

∣∣∣ 6 1 and let usdenote by φ1 and φ2 the solutions of

φ′′1 + (1− k2)φ1 = 0 and φ′′2 + (−1− k2)φ2 = 0,

respectively, with φj(0) = 0 and φ′j(0) = 1, where j = 1, 2. It is straightforward tosee that

φ1(y) =1√

k2 − 1sinh(

√k2 − 1)y and φ2(y) =

1√k2 + 1

sinh(√

k2 + 1)y.

Sturm’s second comparison theorem [25] then asserts that the solution φc,k of (2.15)and φ1 φ2 satisfy that

φ′1φ1

6φ′c,k

φc,k6 φ′2

φ2,

and thus φ1 6 φc,k 6 φ2. It is then easy to see that f(c) > 0 when k is big enough.(b) Suppose on the contrary that c(k1) = c(k2) = c for k1 < k2. Let us denote

by φc,k1 and φc,k2 be the corresponding nontrivial solutions of (2.9)–(2.10). BySturm’s second comparison theorem [25] follows that

φ′c,k1(h)

φc,k1(h)<

φ′c,k2(h)

φc,k2(h).

This contradicts the boundary condition (2.10).(c) The Rayleigh equation (2.13) for k = 0 implies that

(c− U(h))φ′c(h) + U ′(h)φc(h) = m,

where m is a constant. On the other hand, an integration of (2.13) yields that

m =g

(c− U(h))φc(h),

and in turn

φc(h) = (c− U(h))∫ h

0

m

(c− U(y))2dy.

These proves the assertion. ¤

The limiting parameter value µ which corresponds to the limiting wave speedc(0) gives the lowest hydraulic head B defined in (2.6); see [14, Section 3] for detail.The limiting wave speed c(0) is the critical value of parameter near which solitarywaves of elevation exist [27].

3. Linearization of the periodic gravity water-wave problem

This section includes the detailed account of the linearization of the water-wave problem (1.1)–(1.4) around a periodic traveling wave which solves (2.2). Thegrowing-mode problem is formulated as a set of operator equations. Invariants ofthe linearized problem are derived, and their implications in the stability of waterwaves are discussed. Our derivations are free from restrictions on the amplitude ofthe steady solution.


3.1. Derivation of the linearized problem of periodic water waves. A peri-odic traveling-wave solution of (2.2) is held fixed, as such it serves as the undisturbedstate about which the system (1.1)–(1.4) is linearized. The derivation is performedin the moving frame of references, in which the wave profile appears to be stationaryand the flow is steady. Let us denote the undisturbed wave profile and (relative)stream function by ηe(x) and ψe(x, y), respectively, which satisfy the system (2.2).The steady (relative) velocity field (ue(x, y)− c, ve(x, y)) = (ψey(x, y),−ψex(x, y))is given by (2.1), and the hydrostatic pressure Pe(x, y) is determined through (2.4).Let

De = (x, y) : 0 < x < 2π/α , 0 < y < ηe(x) and Se = (x, ηe(x)) : 0 < x < 2π/αdenote, respectively, the undisturbed fluid domain of one period and the steadywave profile. The steady vorticity ωe(x, y) may be expressed as ωe = −∆ψe =γ(ψe).

The linearization concerns a slightly-perturbed time-dependent solution of thenonlinear problem (1.1)–(1.4) near the steady state (ηe(x), ψe(x, y)). Let us denotethe small perturbation of the wave profile, the velocity field and the pressure byηe(x)+η(t; x), (ue(x, y)−c+u(t; x, y), ve(x, y)+v(t; x, y)) and Pe(x, y)+P (t;x, y),respectively. The idea is to expand the nonlinear equations in (1.1)–(1.4) aroundthe steady state in the order of small perturbations and restrict the first-order termsto the unperturbed domain and the steady boundary to obtain linearized equationsfor the the deviations η(t; x), (u(t; x, y), v(t; x, y)), and P (t;x, y) in the wave profile,the velocity field and the pressure from those of the undisturbed state.

In the steady fluid domain De, the velocity deviation (u, v) satisfies the incom-pressibility condition

(3.1) ∂xu + ∂yv = 0

and the linearized Euler equation

(3.2)

∂tu + (ue − c)∂xu + uexu + ve∂yu + veyv = −∂xP

∂tv + (ue − c)∂xv + vexu + ve∂yv + veyv = −∂yP,

where P is the pressure deviation. Equation (3.1) allows us to introduce the streamfunction ψ(t; x, y) for the velocity deviation (u(t;x, y), v(t;x, y)):

∂xψ = −v and ∂yψ = u.

Let us denote by ω(t; x, y) the deviation in vorticity from that of the steady flowωe. By definition, ω = −∆ψ. This motivates us to write (3.2) into the linearizedvorticity equation as

(3.3) ∂tω + (ψey∂xω − ψex∂yω) + (ωex∂yψ − ωey∂xψ) = 0.

Since ωe = γ(ψe), the last term can be written as −γ′(ψe)(ψey∂xψ − ψex∂yψ).The linearized kinematic and dynamic boundary conditions restricted to the

steady free boundary Se are

(3.4) v + veyη = ∂tη + (ue − c)∂xη + (u + ueyη)ηex

andP + Peyη = 0,


respectively. In terms of the stream function, (3.4) is further written as

0 = ∂tη + ψey∂xη + (∂xψ + ηex∂yψ) + (ψexy + ηexψeyy)η= ∂tη + ψey∂xη + ∂τψ + ∂τψeyη

= ∂tη + ∂τ (ψeyη) + ∂τψ,

where∂τf = ∂xf + ηex∂yf

denotes the tangential derivative of a function f defined on the curve y = ηe(x).Alternatively, ∂τf(x) = ∂xf(x, ηe(x)). The bottom boundary condition of thelinearized motion is

∂xψ = 0 on y = 0.Our next task is to examine the time-evolution of ψ on the steady free surface

Se. This links the tangential derivative of the pressure deviation P on the steadyfree surface Se with ψ and η on Se. For a function f defined on Se, let us denoteby

∂nf = ∂yf − ηex∂xf

the normal derivative of f on the curve y = ηe(x).Lemma 3.1. On the steady free surface Se, the normal derivative ∂nψ satisfies

∂t∂nψ + ∂τ (ψey∂nψ) + Ω∂τψ + ∂τP = 0,

where Ω = ωe(x, ηe(x)) is the (constant) value of steady vorticity on Se.

Proof. The linearized Euler equation (3.2) may be rewritten in the vector form as

−∇P = ∂t

(uv

)+∇((ue − c)u + vev)− ωe

(v−u

)− ω

(ve

−ue

)

Restricting the above to the steady free surface Se and computing the tangentialderivative yield that

−∂τP = ∂t(u + ηexv) + ∂τ ((ue − c)u + vev)− Ω(v − ηexu)

= ∂t(∂yψ − ηex∂xψ) + ∂τ (ψey∂yψ − ψeyηex∂xψ)− Ω(−∂xψ − ηex∂yψ)

= ∂t∂nψ + ∂τ (ψey∂nψ) + Ω∂τψ.

The second equality uses the kinematic boundary condition where in the abovederivation we use the steady kinematic equation for the steady state ψex = ψeyηex

on Se. ¤In summary, there results in the linearized water-wave problem:

∂tω + (ψey∂xω − ψex∂yω) = γ′(ψe)(ψey∂xψ − ψex∂yψ) in De,(3.5a)

where ω = −∆ψ;

∂tη + ∂τ (ψeyη) + ∂τψ = 0 on Se;(3.5b)

P + Peyη = 0 on Se;(3.5c)

∂t∂nψ + ∂τ (ψey∂nψ) + Ω∂τψ + ∂τP = 0 on Se;(3.5d)

∂xψ = 0 on y = 0.(3.5e)

Note that the above linearized system may be viewed as one for ψ(t; x, y) andη(t; x). Indeed, P (t;x, ηe(x)) is determined through (3.5c) in terms of η(t; x) andother physical quantities are similarly determined in terms of ψ(t; x, y) and η(t; x).


3.2. The growing-mode problem. A growing mode refers to a solution to thelinearized water-wave problem (3.5) of the form

(η(t; x), ψ(t;x, y)) = (eλtη(x), eλtψ(x, y))

and P (t; x, ηe(x)) = eλtP (x, ηe(x)) with Re λ > 0. For such a solution, the lin-earized vorticity equation (3.5a) further reduces to

(3.6) λω + (ψey∂xω − ψex∂yω)− γ′(ψe)(ψey∂xψ − ψex∂yψ) = 0 in De,

where ω = −∆ψ. Let (Xe(s; x, y), Ye(s;x, y)) be the particle trajectory of thesteady flow

(3.7)

Xe = ψey(Xe, Ye)Ye = −ψex(Xe, Ye)

with the initial position (Xe(0), Ye(0)) = (x, y). Here, the dot above a variabledenotes the differentiation in the s-variable. Integration of (3.6) along the particletrajectory (Xe(s;x, y), Ye(s, x, y)) for s ∈ (−∞, 0) yields [34, Lemma 3.1]

(3.8a) ∆ψ + γ′(ψe)ψ − γ′(ψe)∫ 0

−∞λeλsψ(Xe(s), Ye(s))ds = 0 in De.

For a growing mode the boundary conditions (3.5b), (3.5c) and (3.5d) on Se

become

λη(x) +d

dx

(ψey(x, ηe(x))η(x)

)= − d

dxψ(x, ηe(x)),(3.8b)

P (x, ηe(x)) + Pey(x, ηe(x))η(x) = 0,(3.8c)

λψn(x) +d

dx

(ψey(x, ηe(x))ψn(x)

)= − d

dxP (x, ηe(x))− Ω

d

dxψ(x, ηe(x)).(3.8d)

The kinematic boundary condition (3.5e) at the flat bottom y = 0 says thatψ(x, 0) is a constant. Observe that (3.8a)-(3.8d) remain unchanged by adding aconstant on ψ, whereby the boundary condition for a growing mode at the bottomy = 0 reduces to

(3.8e) ψ(x, 0) = 0.

In summary, the growing-mode problem for periodic traveling water-waves is tofind a nontrivial solution of (3.8a)-(3.8e) with Re λ > 0.

3.3. Invariants of the linearized water-wave problem. In this subsection, theinvariants of the linearized water-wave problem (3.5a)–(3.5e) are derived, and theirimplications in the stability of water waves are discussed.

With the introduction of the Poisson bracket, defined as

[f1, f2] = ∂xf1∂yf2 − ∂yf1∂xf2,

the linearized vorticity equation (3.5a) is further written as

(3.9) ∂tω − [ψe, ω] + γ′(ψe)[ψe, ψ] = 0 in De,

where ω = −∆ψ. Recall that

∂τf = ∂xf + ηex∂yf and ∂nf = ∂yf − ηex∂xf,

where f is a function defined on Se.Our first task is to find the energy functional for the linearized problem which,

in case that the vorticity-stream function relation is monotone, is an invariant.


Lemma 3.2. Provided that either γ′(p) < 0 or γ′(p) > 0 on [0, |p0|], then for anysolution (η, ψ) of the linearized water-wave problem (3.5), we have d

dtE(η, ψ) = 0,where

E(η, ψ) =∫∫

De

12 |∇ψ|2dydx−

∫∫

De

12γ′(ψe)−1|ω|2dydx

−∫

Se

12Pey|η|2dx + Re

∫

Se

ψey∂nψη∗dx− Ω∫

Se

12ψey|η|2dx.

(3.10)

In the irrotational setting, i.e. γ ≡ 0, the invariant functional reduces to

(3.11) E(η, ψ) =∫∫

De

12 |∇ψ|2dydx−

∫

Se

12Pey|η|2dx + Re

∫

Se

ψey∂nψη∗dx.

Proof. In view of the divergence theorem it follows thatd

dt

∫∫

De

12 |∇ψ|2dydx = Re

∫∫

De

∂t(∇ψ) · ∇ψ∗dydx

= Re∫∫

De

∂tωψ∗dydx +∫

Se

∂t(∂nψ)ψ∗dx +∫

y=0∂t∂yψψ∗dx

:= (I) + (II) + (III).

A substitution of ∂tω by the linearized vorticity equation (3.9) yields that

(I) = Re∫∫

De

(−γ′(ψe)[ψe, ψ] + [ψe, ω]) ψ∗dydx

= Re∫∫

De

(− 12 [ψe, γ

′(ψe)|ψ|2] + [ψe, ωψ∗]− [ψe, ψ∗]ω

)dxdy.

The second equality in the above uses that12 [ψe,γ

′(ψe)|ψ|2]= 1

2 [ψe, γ′(ψe)]|ψ|2 + 1

2γ′(ψe)([ψe, ψ]ψ∗ + [ψe, ψ∗]ψ)

= γ′(ψe) Re[ψe, ψ]ψ∗,

which follows since [ψe, f(ψe)] = 0 for any f . With the observation that∫∫

De

[ψe, f ]dydx =∫∫

De

∇ · ((ψey,−ψex)f)dydx

=∫

Se

(ψey,−ψex) · (−ηex, 1)fdx +∫

y=0

fψexdx = 0

for any function f , the integral (I) further reduces to

(I) = Re∫∫

De

−[ψe, ψ]∗ωdydx.

A simple substitution of [ψe, ψ] by the linearized vorticity equation (3.9) then yieldsthat

(I) = Re∫∫

De

γ′(ψe)−1(∂tω∗ − [ψe, ω]∗)ωdydx

= Re∫∫

De

( 12γ′(ψe)−1∂t|ω|2 − 1

2 [ψe, γ′(ψe)−1|ω|2])dydx(3.12)

=d

dt

∫∫

De

12γ′(ψe)−1|ω|2dydx.


This uses that12 [ψe, γ

′(ψe)−1|ω|2] = γ′(ψe)−1 Re[ψe, ω]∗ω.

Next is to examine the surface integral (II). Simple substitutions by the lin-earized boundary conditions (3.5b), (3.5c) and (3.5d) into (II) and an integrationby parts yield that

(II) = Re∫

Se

∂τ (Peyη − ψey∂nψ − Ωψ)ψ∗dx

= Re∫

Se

(Peyη − ψey∂nψ)(−∂τψ∗)dx

= Re∫

Se

(Peyη − ψey∂nψ)(∂tη∗ + ∂τ (ψeyη∗))dx

=d

dt

∫

Se

12Pey|η|2dx− Re

∫

Se

ψey∂nψ∂tη∗dx + Re

∫

Se

∂τ (P + ψey∂nψ)ψeyη∗dx.

The second equality uses that

Re∫

Se

(∂τψ)ψ∗dx =12

∫

Se

d

dx|ψ(x, ηe(x))|2 dx = 0.

More generally, Re∫Se

(∂τf)f∗dx = 0 for any function f defined on Se. Withanother simple substitution by the boundary condition (3.5d), the last term in thecomputation of (II) is written as

Re∫

Se

∂τ (P + ψey∂nψ)ψeyη∗dx

= −Re∫

Se

(∂t∂nψ + Ω∂τψ)ψeyη∗dx

= −Re∫

Se

(∂t(∂nψ)ψeyη∗dx + Ω Re∫

Se

(∂tη + ∂τ (ψeyη))ψeyη∗)dx

= −Re∫

Se

ψey∂t(∂nψ)η∗dx + Ωd

dt

∫

Se

12ψey|η|2dx.

The last equality uses that Re∫Se

∂τ (ψeyη)(ψeyη)∗dx = 0. Therefore,

(3.13) (II) =d

dt

∫

Se

12Pey|η|2dx− Re

d

dt

∫

Se

ψey∂nψη∗dx + Ωd

dt

∫

Se

12ψey|η|2dx.

Finally, it is straightforward to see that

(III) = ψ∗(x, 0)∫

y=0∂t(∂yψ)dx = 0.

This, together with (3.12) and (3.13) asserts that E(η, ψ) is an invariant. In theirrotational setting, i.e. γ = 0, the area integral (I) is zero, Ω = 0, and the otherterms remain the same. This completes the proof. ¤

Remark 3.3. Our energy functional E agrees with the second variation ∂2H of theenergy-Casimir functional in [15]. Recall that the hydrostatic pressure of the steadysolution is given in (2.4) as

Pe(x, y) = B − 12 |∇ψe(x, y)|2 − gy +

∫ ψe(x,y)

0

γ(−p)dp,


where B is the Bernoulli constant. Differentiation of the above and restriction onSe then yield that

−Pey − Ωψey = 12∂y|∇ψe|2 + g,

where Ω = γ(ψ = 0) = ωe(x, ηe(x)). Thus, in case γ is monotone, it follows that

2E(η, ψ) =∫∫

De

|∇ψ|2dydx−∫∫

De

γ′(ψe)−1|ω|2dydx

+∫

Se

(12∂y|∇ψe|2 + g

) |η|2dx + 2 Re∫

Se

ψey∂nψη∗dx.

This is exactly the expression of the second variation ∂2H in [15] of the enery-Casimir functional, which in our notations

(3.14) H(η, ψ) =∫∫

Dη

( |∇(ψ − cy)|22

+ gy −B − F (ω))

dydx,

around the steady state (ηe, ψe). Here, Dη = (x, y) : 0 < y < η(t; x) and(F ′)−1 = γ. The quadratic form ∂2H is used in [15] to study formal stability.

Our next invariant is the linearized horizontal momentum. The result is freefrom restrictions on γ.

Lemma 3.4. For any solution (η, ψ) of the linearized problem of (3.5), the identity

d

dt

∫

Se

(ψ + ψeyη)dx = 0

holds true.

Proof. We integrate over the steady fluid region De of the linearized equation forthe horizontal velocity

∂t∂yψ + (ψey,−ψex) · ∇(∂yψ) + (∂yψ,−∂xψ) · ∇ψey = −Px

and apply the divergence theorem to arrive at

(3.15)d

dt

∫∫

De

∂yψ dydx +∫

Se

(∂yψ,−∂xψ) · (−ηex, 1)ψeydx = −∫∫

De

Pxdydx.

It is straightforward to see that∫∫

De

∂yψdydx =∫

Se

ψ dx.

In view of (3.5b), the second term on the left hand side of (3.15) is written as∫

Se

(∂τψ)ψeydx =∫

Se

(∂tη + ∂τ (ψeyη))ψeydx

=d

dt

∫

Se

ψeyη dx−∫

Se

ψey∂τ (ψey)ηdx.

With the use of Stokes’ theorem and the dynamic boundary condition (3.1) theright side of (3.15) becomes

∫∫

De

Pxdydx =∫

Se

Pηexdx =∫

Se

Peyηηexdx.


On the other hand, the steady Euler equation restricted to the steady free-surfaceSe yields that

−Pex = ψeyψexy − ψexψeyy

= ψey(ψexy + ηexψeyy) = ψey∂τψey

Since Pex + Peyηex = 0 on Se, a simple substitution then proves the assertion. ¤

Next, integration of (3.5b) on Se also yields that∫Se

η dx is invariant. Finally,multiplication on the linearized vorticity equation (3.9) by ξ and then integrationyield that

∫∫

De

ωξdydx

is an invariant, for any function

ξ ∈ ker(ψey∂x − ψex∂y) ⊂ L2(De).

We summarize our results.

Proposition 3.5. The linearized problem (3.5) has the energy invariant:

E(η, ψ) =∫∫

De

12 |∇ψ|2dydx−

∫∫

De

12γ′(ψe)−1|ω|2dydx

+∫

Se

12

(∂y( 1

2 |∇ψe|2) + g) |η|2dx + Re

∫

Se

ψey∂nψη∗dx

if γ is monotone and

E(η, ψ) =∫∫

De

12 |∇ψ|2dydx

+∫

Se

12

(∂y( 1

2 |∇ψe|2) + g) |η|2dx + Re

∫

Se

ψey∂nψη∗dx

in case γ ≡ 0. In addition, (3.5) has the following invariants:

M(η, ψ) =∫

Se

(ψ + ψeyη)dx,

m(η, ψ) =∫

Se

ηdx

F(η, ψ) =∫∫

De

ωξdydx

for any ξ ∈ ker(ψey∂x − ψex∂y).

The nonlinear water-wave problem (1.1)–(1.4) has the following conservationlaws [15, Section 2]: Let

D(t) = (x, y) : 0 < x < 2π/α , 0 < y < η(t; x)


be the fluid domain at time t of one wave length, and

E =∫∫

D(t)

(12 (u2 + v2) + gy

)dydx (energy),

M =∫∫

D(t)

u dydx (horizontal momentum),

m =∫∫

D(t)

dydx (mass),

F =∫∫

D(t)

f(ω)dydx (Casimir invariant),

where the function f is arbitrary such that the integral F exists.The invariants of the linearized problem E , M, m, and F in Proposition 3.5 can

be obtained by expanding the invariants of the nonlinear problem E, M, m and F,respectively, around the steady wave (ηe(x), ψe(x, y)). The quadratic form E(η, ψ)is the second variation of the energy functional H given in (3.14) as is shown inRemark 3.3, which is a combination of E, M, m and F. The invariants M and m ofthe linear problem are the first variations of M and m, respectively. Finally, F is thefirst variation of F sinceby the assumption of no stagnation and the monotonicityassumption it follows that

ker(ψey∂x − ψex∂y) = f(ψe) : f is arbitrary = f(ωe) : f is arbitrary.We now make some comments on the implications of these invariants on the

stability of water waves. A traditional approach to the stability of conservativesystems is the so-called energy method, for which one tries to show that a steadystate is an energy minimizer under the constraints of other invariants such as mo-mentum, mass, etc. This method has been widely used in the stability analysis ofvarious approximate equations such as the KdV equation [5], [8] and the water-waveproblem with a nonzero coefficient of surface tension [38]. Nonetheless, a steadysolution of the fully-nonlinear gravity water-wave problem in general is not expectto be an energy minimizer, as is discussed below. Note that if a steady gravitywater-wave is an energy minimizer under constraints of fixed M, m and F, then inthe linearized level the second variation E should be positive under the constraintsthat the variations M, m and F are zero.

In the irrotational setting, the first two terms in the expression (3.11) of E(η, ψ)yield some positive contribution since they are equivalent to

(3.16)∫∫

De

|∇ψ|2dydx +∫

Se

|η|2dx.

(Indeed, since ∆Pe = −2ψ2exy − ψ2

exx − ψ2eyy 6 0 and Pey(x, 0) = −g by the

maximum principle Pe attains its minimum at the free surface Se. Furthermore,since Pe takes a constant on Se by the Hopf lemma Pey < 0 on Se.) The last termRe

∫Se

ψey∂nψη∗dx of the right side of (3.11), however, does not have a definite signand contains a 1/2-higher derivative than that of (3.16). Thus, it cannot be boundedby elements in the positive contribution (3.16), regardless of the constraints M =m = 0. Consequently, the quadratic form E(η, ψ) is indefinite unless ψey ≡ 0, thatis, the steady flow is uniform and the steady surface is flat. In other words, a steadygravity water-wave in the irrotational setting is in general not an energy minimizer.


Indeed, in [23], [11], steady water-waves were constructed as an energy saddle bymeans of variational methods.

With a nonzero vorticity, a control of the mixed-type term in the energy func-tional by other positive terms fails for the same reason as in the irrotational setting,even for a vorticity which would amplify positivity property. Let us consider a vor-ticity function which decreases with the flux, that is, γ′ < 0. Under some additionalassumptions, it is shown in [15] that the first three terms in the expression (3.10)of E(η, ψ) give rise a positive norm

(3.17)∫∫

De

(|∇ψ|2 + |ω|2)dydx +∫

Se

|η|2dx.

A successful bound of the mixed-type term Re∫Se

ψey∂nψη∗dx by terms in (3.17)would entail a control of ψ on Se in terms of positive contribution in (3.17), whichdoes not seem to follow from the consideration of elliptic regularity. In [15], severalclasses of perturbations were introduced to make the mixed-type term controllableby the elements in (3.17) and thus to ensure the positivity of the energy E(η, ψ) inthese classes; The formal stability of the first kind in this regard corresponds to thepositivity of E(η, ψ). However, it is difficult to show that these special classes areinvariant under the evolution process of the water-wave problem, and it remainsunclear how to pass from the formal stability of the first kind to the genuine stabilityeven under some special perturbations. The lack of a control of the mixed-type termin the energy expression by other positive terms is not amendable by taking theother invariants M,m, and F into consideration as constraints while one may relaxthe assumptions on the vorticity to obtain the positivity of (3.17). In conclusion,the quadratic form E(η, ψ) is in general indefinite in the rotational setting, androtational steady water-wave is again expected to be an energy saddle.

The above discussions of steady water-waves as energy saddles do not implythat steady water-waves are necessarily unstable. Indeed, as mentioned before,the small Stokes waves are believed to be stable [39], [45] under perturbations ofthe same period. For the rotational case, under the assumption of a monotoneγ, the corresponding trivial solutions with shear flows defined in Lemma 2.1 cannot have an inflection point since γ′(ψe(y)) = −U ′′(y)/U(y) 6= 0 and the no-stagnation assumption ensures that U < 0. Thus by Theorem 6.4, such shear flowsare linearly stable to perturbations of any period. Since small-amplitude waveswith any monotone vorticity relation γ bifurcate from these strongly stable shearflows, they are likely to be stable. But, a successful stability analysis of generalsteady gravity water-waves would require to use the full set of equations instead ofa few of invariants.

4. Linear instability of shear flows with free surface

This section is devoted to the study of the linear instability of a free-surface shearflow (U(y), 0) in y ∈ [0, h], a steady solution of the water-wave problem (1.1)–(1.4)with P (x, y) = −gy. In the frame of reference moving with the speed c > max U ,this may be recognized as a trivial solution of the traveling-wave problem (2.2):

ηe(x) ≡ h and (ψey(x, y),−ψex(x, y)) = (U(y)− c, 0).

Throughout this section, we write U for U − c for simplicity of the presentation.


We seek for normal mode solutions of the form

η(t;x) = ηheiα(x−ct), ψ(t; x, y) = φ(y)eiα(x−ct)

and P (t; x, y) = P (y)eiα(x−ct) to the linearized water-wave problem (3.5). Here,α > 0 is the wave number and c is the complex phase speed. It is equivalent to findsolutions to the growing-mode problem (3.8) of the form λ = −iαc and

η(x) = ηheiαx, ψ(x, y) = φ(y)eiαx

and P (x, ηe(x)) = Pheiαx. Note that Re λ > 0 if and only if Im c > 0.Since γ′(ψe(y)) = ωey(y)/ψey(y) = −U ′′(y)/U(y) and (Xe(s), Ye(s)) = (x +

U(y)s, y), the linearized vorticity equation (3.8a) translates into the Rayleigh equa-tion

(4.1) (U − c)(φ′′ − α2φ)− U ′′φ = 0 for y ∈ (0, h).

Here and elsewhere the prime denote the differentiation in the y-variable. Theboundary conditions (3.8b), (3.8c) and (3.8d) on the free surface are simplified tobe

(c− U(h))ηh = φ(h), Ph − gηh = 0and

(c− U(h))φ′(h) = Ph + U ′(h)φ(h),respectively. Eliminating ηh and Ph from the above yields that

(U(h)− c)2φ′(h) = (g + U ′(h)(U(h)− c)) φ(h).

The bottom boundary condition (3.8e) becomes φ(0) = 0. In summary, there resultsin the Rayleigh equation (4.1) with the boundary condition

(4.2)

(U(h)− c)2φ′(h) = (g + U ′(h)(U(h)− c)) φ(h)φ(0) = 0.

A shear profile U is said to be linearly unstable if there exists a nontrivial solutionof (4.1)-(4.2) with Im c > 0.

The Rayleigh system (4.1)–(4.2) is alternatively derived in [48] by linearizingdirectly the water-wave problem (1.1)–(1.4) around (U(y), 0) in y ∈ [0, h]. Notethat with c > maxU the Rayleigh system (4.1)–(4.2) is the bifurcation equation(2.9)–(2.10).

In case of rigid walls at y = h and y = 0, that is the Dirichlet boundary conditionsφ(h) = 0 = φ(0) in place of (4.2), the instability of a shear flow is a classical problem,which has been under extensive research since Lord Rayleigh [42]. Recently, by anovel analysis of neutral modes, Lin [32] established a sharp criterion for linearinstability in the rigid-wall setting of a general class of shear flows. Our objectivein this section is to obtain an analogous result in the free-surface setting.

Below is the definition of the class of shear flows studied in this section. By aninflection value we mean the value of U at an inflection point.

Definition 4.1. A function U ∈ C2([0, h]) is said to be in the class K+ if U has aunique inflection value Us and

(4.3) K(y) = − U ′′(y)U(y)− Us

is bounded and positive on [0, h].


Typical example of K+-flows include U(y) = cos my and U(y) = sin my.For U ∈ K+ let us consider the Sturm-Liouville equation

(4.4) φ′′ − α2φ + K(y)φ = 0 for y ∈ (0, h)

with the boundary conditionsφ′(h) = gr(Us)φ(h) if U(h) 6= Us

φ(h) = 0 if U(h) = Us,(4.5)

φ(0) = 0.(4.6)

Here,

(4.7) gr(c) =g

(U(h)− c)2+

U ′(h)U(h)− c

.

The following theorem states a sharp instability criterion of free-surface gravityshear-flows in class K+.

Theorem 4.2 (Linear instability of free-surface shear flows in K+). For U ∈K+, let the lowest eigenvalue −α2

max of the ordinary differential operator − d2

dy2 −K(y) on the interval y ∈ (0, h) with the boundary conditions (4.5)–(4.6) exists andbe negative. Then, to each α ∈ (0, αmax) corresponds an unstable solution-triple(φ, α, c) (with Im c > 0) of (4.1)–(4.2). The interval of unstable wave numbers(0, αmax) is maximal in the sense that the flow is linearly stable if either − d2

dy2−K(y)on y ∈ (0, h) with (4.5)–(4.6) is nonnegative or α > αmax.

From a variational consideration, the lowest eigenvalue −α2max is characterized

as

(4.8a) −α2max = inf

φ∈H1(0,h)φ(0)=0

∫ h

0(|φ′|2 −K(y)|φ|2)dy + gr(Us)|φ(h)|2

∫ h

0|φ|2dy

in case U(h) 6= Us, and

(4.8b) −α2max = inf

φ∈H1(0,h)φ(0)=0=φ(h)

∫ h

0(|φ′|2 −K(y)|φ|2)dy

∫ h

0|φ|2dy

in case U(h) = Us.

4.1. Neutral limiting modes. The proof of Theorem 4.2 makes use of neutrallimiting modes, as in the rigid-wall setting [32].

Definition 4.3 (Neutral limiting modes). A triple (φs, αs, cs) with αs positive andcs real is called a neutral limiting mode if it is the limit of a sequence of unstablesolutions (φk, αk, ck)∞k=1 of (4.1)–(4.2) as k → ∞. The convergence of φk to φs

is in the almost-everywhere sense. For a neutral limiting mode, αs is called theneutral limiting wave number and cs is called the neutral limiting wave speed.

Lemma 4.8 below will establish that neutral limiting wave numbers form theboundary points of the interval of unstable wave numbers, and thus the stabil-ity investigation of a shear flow reduces to finding all neutral limiting modes andstudying stability properties near them. In general, it is difficult to locate all neu-tral limiting modes. For flows in class K+, nonetheless, neutral limiting modes arecharacterized by the inflection value.


Proposition 4.4 (Characterization of neutral limiting modes). For U ∈ K+ aneutral limiting mode (φs, αs, cs) must solve (4.4)–(4.6) with cs = Us.

For the proof of Proposition 4.4, we need several properties of unstable solu-tions. First, Howard’s semicircle theorem holds true in the free-surface setting [48,Theorem 1]. That is, any unstable eigenvalue c = cr + ici (cr > 0) of the Rayleighequation (4.1)-(4.2) must lie in the semicircle

(4.9) (cr − 12 (Umin + Umax))2 + c2

i 6 ( 12 (Umin − Umax))2,

where Umin = min[0,h] U(y) and Umax = max[0,h] U(y).The identities below are useful for future considerations.

Lemma 4.5. If φ is a solution (4.1)–(4.2) with c = cr + ici and ci 6= 0 then forany q real the identities

∫ h

0

(|φ′|2 + α2|φ|2 +

U ′′(U − q)|U − c|2 |φ|2

)dy =

(Re gr(c) +

cr − q

ciIm gr(c)

)|φ(h)|2,

(4.10)

∫ h

0

(|φ′|2 + α2|φ|2 +

U ′′(U − q)|U − c|2 |φ|2

)dy =

(Re gs(c)− cr − q

ciIm gs(c)

)|φ′ (h)|2

(4.11)

hold true, where gr is defined in (4.7) and

(4.12) gs(c) =(U(h)− c)2

g + U ′(h)(U(h)− c).

Proof. Multiplication of (4.1) by φ∗ and integration by parts using the boundarycondition (4.2) yield

∫ h

0

(|φ′|2 + α2|φ|2 +

U ′′

U − c|φ|2

)dy = gr(c)|φ(h)|2.

Its real and imaginary parts read as

(4.13)∫ d

0

(|φ′|2 + α2 |φ|2 +

U ′′ (U − cr)|U − c|2 |φ|2

)dy = Re gr(c) |φ (d)|2

and

(4.14) ci

∫ h

0

U ′′

|U − c|2 |φ|2dy = Im gr(c)|φ(h)|2,

respectively. Combining (4.13) and (4.14) then establishes (4.10).Similarly, combining the real and the imaginary parts of

(4.15)∫ h

0

(|φ′|2 + α2|φ|2 +

U ′′

U − c|φ|2

)dy = gs(c)|φ′(h)|2

leads to (4.11). This completes the proof. ¤Note that (4.14) is written as

ci

∫ h

0

U ′′

|U − c|2 |φ|2dy =

(2g(U(h)− cr)|U(h)− c|4 +

U ′(h)|U(h)− c|2

)|φ(h)|2,

and hence, if U ′′ does not change sign and U is monotone then the flow is linearlystable [48, Section 5]. Proposition 6.4 will prove in the free-surface setting linear


stability of general class of shear flows with no inflection points (and U need notbe monotone). The proof uses the characterization of neutral limiting modes.

Our next preliminary result is an a priori H2-estimate of unstable solutions neara neutral limiting mode.

Lemma 4.6. For U ∈ K+ let (φk, αk, ck)∞k=1 be a sequence of unstable solutionsto (4.1)–(4.2) such that ‖φk‖L2 = 1. If αk → αs > 0 and Im ck → 0+ as k → ∞then ‖φk‖H2 6 C, where C > 0 is independent of k.

Proof. By the semicircle theorem (4.9), it follows that ck → cs and cs ∈ [Umin, Umax]as k →∞. The proof is divided into to cases when U(h) 6= cs and when U(h) = cs.

Case 1: U(h) 6= cs. The proof is similar to that of [32, Lemma 3.7]. It isstraightforward to see that

|Re gr(ck)|+∣∣∣∣Im gr(ck)

Im ck

∣∣∣∣ 6 C0,

where C0 > 0 is independent of k. For simplicity, the subscript k is suppressed inthe estimate below and C is used to denote generic constants which are independentof k. Let c = cr + ici. Let us write (4.10) as∫ h

0

(|φ′|2 + α2|φ|2 + K(y)

(U − Us)(U − q)|U − cr|2 + c2

i

|φ|2)

dy =(

Re gr(c) +cr − q

ciIm gr(c)

)|φ(h)|2 .

Note that gr(cs) is well defined. Setting q = Us − 2 (Us − cr) the above identityleads to∫ h

0

(|φ′|2 + α2|φ|2)dy 6∫ h

0

K(y)(U − Us)2 + 2(U − Us)(Us − cr)

|U − cr|2 + c2i

|φ|2dy + C|φ(h)|2

=∫ h

0

K(y)(U − cr)2 − (Us − cr)2

|U − cr|2 + c2i

|φ|2dy + C|φ(h)|2

6∫ h

0

K(y)|φ|2dy + C(ε ‖φ′‖2L2 +1ε‖φ‖2L2).

One chooses ε sufficiently small to conclude that ‖φ‖H1 6 C.Next is an H2-estimate. Multiplication of (4.1) by (φ∗)′′ and integration over

the interval [0, h] yield

(4.16)∫ h

0

(|φ′′|2 + α2 |φ′|2)dy − α2g∗r (c)|φ(h)|2 =∫ h

0

(φ∗)′′U ′′

U − cφ dy.

In view of the Rayleigh equation for φ∗, the right side is written as∫ h

0

(φ∗)′′U ′′

U − cφ dy =

∫ h

0

(α2φ∗ +

(U ′′

U − cφ

)∗)U ′′

U − cφ dy

= α2

∫ h

0

U ′′

U − c|φ|2dy +

∫ h

0

(U ′′)2

|U − c|2 |φ|2dy.

The real part of (4.16) then reads as∫ h

0

(|φ′′|2 + α2 |φ′|2)dy = α2 Re gr(c) |φ(h)|2 + α2

∫ h

0

U ′′(U − cr)|U − c|2 |φ|2dy +

∫ h

0

(U ′′)2

|U − c|2 |φ|2dy

= α2

(2Re gr(c)|φ(h)|2 −

∫ h

0

(|φ′|2 + α2|φ|2)dy

)+

∫ h

0

(U ′′)2

|U − c|2 |φ|2dy.


The last equality uses (4.13). On the other hand, (4.10) with q = Us implies∫ h

0

(U ′′)2

|U − c|2 |φ|2dy 6 ‖K‖L∞

∫ h

0

−U ′′(U − Us)|U − c|2 |φ|2dy

= ‖K‖L∞

(∫ h

0

(|φ′|2 + α2|φ|2)dy −(

Re gr(c) +cr − Us

ciIm gr(c)

)|φ(h)|2

)

6 C ‖φ‖2H1 .

Since |Re gr(ck)||φ(h)|2 6 C0‖φ‖2H1 , a simple substitution results in∫ h

0

(|φ′′|2+2α2 |φ′|2+α4|φ|2)dy = −2α2 Re gr(c)|φ(h)|2+∫ h

0

(U ′′)2

|U − c|2 |φ|2dy 6 C ‖φ‖2H1 .

Therefore, ‖φ‖H2 6 C as desired.Case 2: U(h) = cs. The proof is identical to that of Case 1 except that one uses

φ(h) = gs(c)φ′(h) in place of φ′(h) = gr(c)φ(h). It is straightforward to see that

|Re gs(ck)|+∣∣∣∣Im gs(ck)

Im ck

∣∣∣∣ 6 C0 d(ck, U(h)),

where C0 > 0 is independent of k and d is defined as

d(ck, U(h)) = |Re ck − U(h)|+ (Im ck)2 .

Since U(h) = cs follows that d(ck, U(h)) → 0 as k → ∞. The same computationsas in Case 1 establish

∫ h

0

(|φ′k|2 + α2|φk|2)dy 6∫ h

0

K(y)|φk|2dy + C0d(ck, U(h)) |φ′k (h)|2

and

∫ h

0

(|φ′′k |2 + 2α2 |φ′k|2 + α4|φk|2)dy = −2α2 Re gs(c)|φ′k(h)|2 +∫ h

0

(U ′′)2

|U − c|2 |φk|2dy

6 C

(∫ h

0

(|φ′k|2 + α2|φk|2)dy + d(ck, U(h)) |φ′k (h)|2)

,

(4.17)

where C > 0 is independent of k. Consequently,

‖φk‖2H2 6 C1(‖φk‖2L2 + d(ck, U(h)) |φ′k (h)|2) 6 C2(‖φk‖2L2 + d(ck, U(h))‖φk‖2H2),

where C1, C2 > 0 are independent of k. Since d(ck, U(h)) → 0 as k →∞ it followsthat ‖φk‖2H2 6 C. This completes the proof. ¤

For U ∈ K+, if c is in the range of U then U(y) = c holds at a finite numberof points [32, Remark 3.2], denoted by y1 < y2 < · · · < ymc . For convenience, sety0 = 0 and ymc+1 = h. We state our last preliminary result.

Lemma 4.7 ([32], Lemma 3.5). Let φ satisfy (4.1) with α positive and c in therange of U and let U(y) = c for y ∈ y1, y2, . . . , ymc. If φ is sectionally continuouson the open intervals (yj , yj+1), j = 0, 1, . . . ,mc, then φ cannot vanish at bothendpoints of any intervals (yj , yj+1) unless it vanishes identically on that interval.


Proof of Proposition 4.4. Let (φs, αs, cs) be a neutral limiting mode with αs > 0and cs ∈ [Umin, Umax], and let (φk, αk, ck)∞k=1 be a sequence of unstable solutionsof (4.1)–(4.2) such that (φk, αk, ck) converges to (φs, αs, cs) as k → ∞. Afternormalization, we may assume that ‖φk‖L2 = 1.

First, the result of Lemma 4.6 says that ‖φk‖H2 6 C, where C > 0 is independentof k. Consequently, φk converges to φs weakly in H2 and strongly in H1. Then‖φs‖H2 6 C and ‖φs‖L2 = 1. Let y1, y2, · · · , yms

be the roots of U(y)− cs and letS0 be the complement of the set of points y1, y2, . . . , yms

in the interval [0, h].Since φk converges to φs uniformly in C2 on any compact subset of S0, it followsthat φ′′s exists on S0. Since (U − ck)−1, φk and their derivatives up to second orderare uniformly bounded on any compact subset of S0, it follows that φs satisfies

(4.18) φ′′s − α2sφs − U ′′

U − csφs = 0

almost everywhere on [0, h]. Moreover,

(4.19) φ′s(h) =(

g

(U(h)− cs)2+

U ′(h)U(h)− cs

)φs(h) and φs(0) = 0

in case U(h) 6= cs, and

(4.20) φs(h) = 0 and φs(0) = 0

in case U(h) = cs.Next, we claim that cs is the inflection value Us. By Definition 4.1, U ′′(yj) =

−K(yj)(cs − Us) has the same sign for j = 1, · · · ,ms, say positive. Let

Eδ = ∪msi=1y ∈ [0, h] : |y − yj | < δ.

Clearly, Ecδ ⊂ S0 and U ′′(y) > 0 for y ∈ Eδ when δ > 0 sufficiently small. The

proof is again divided into two cases.Case 1: U(h) 6= cs. For any q real

∫ h

0

(|φ′k|2 + α2

k|φk|2 +U ′′(U − q)|U − ck|2 |φk|2

)dy > α2

k+∫

Eδ

U ′′(U − q)|U − ck|2 |φk|2dy

− supEc

δ

|U ′′(U − q)||U − ck|2 .

Since φs is not identically zero, Lemma 4.7 asserts that φs(yj) 6= 0 for some yj ∈ Eδ.If cs were not an inflection value then near such a yj it must hold that

∫

Eδ

U ′′(U − Umin + 1)|U − cs|2 |φs|2dy >

∫

|y−yj |<δ

U ′′

|U − cs|2 |φs|2dy = ∞.

Subsequently, by Fatou’s Lemma it follows that

lim infk→∞

∫ h

0

(|φ′k|2 + α2

k|φk|2 +U ′′(U − Umin + 1)

|U − ck|2 |φk|2)

dy = ∞.

On the other hand, (4.10) with q = Umin − 1 yields∫ h

0

(|φ′′k |2 + α2

k|φk|2 +U ′′(U − Umin + 1)

|U − ck|2 |φk|2)

dy

=(

Re gr(ck) +Re ck − Umin + 1

Im ck

)|φk(h)|2 6 C‖φk‖H2 6 C1,


where C1 > 0 is independent of k. A contradiction proves the claim.Case 2: U(h) = cs. The proof is identical to that of Case 1 except that we use

(4.11) in place of (4.10) and hence is omitted. This completes the proof. ¤

The following lemma permits us to continue unstable wave numbers until itreaches a neutral limiting wave number, analogous to [32, Theorem 3.9].

Lemma 4.8. For U ∈ K+, the set of unstable wave numbers is open, whose bound-ary point α satisfies that −α2 is an eigenvalue of the operator − d2

dy2 − K(y) ony ∈ (0, h) with the boundary conditions (4.5)–(4.6).

Proof. An unstable solution (φ0, α0, c0) of (4.1)–(4.2) is held fixed, and such α0 > 0and Im c0 > 0. For r1, r2 > 0 let us define

Ir1 = α > 0 : |α− α0| < r1 and Br2 = c ∈ C+ : |c− c0| < r2,where C+ denotes the set of complex numbers with positive imaginary part. Ourgoal is to show that for each α ∈ Ir1 , there exists c(α) ∈ Br2 for some r2 > 0 andφα ∈ H2 such that (φα, α, c(α)) satisfies (4.1)–(4.2).

For α ∈ (0,∞) and c ∈ C+, let φ1 (x; α, c) and φ2 (x; α, c) to be the solutions ofthe differential equation

φ′′ − α2φ− U ′′

U − cφ = 0 for y ∈ (0, h)

normalized at h, that is,

φ1(h) = 1, φ2(h) = 0,

φ′1(h) = 0, φ′2(h) = 1.

It is standard that φ1 and φ2 are analytic as a function of c in C+. Let us considera function on (0,∞)× C+, defined as

(4.21) Φ(α, c) = φ1(0; α, c) + gr(c)φ2(0; α, c),

where gr is defined in (4.7). Clearly, Φ is analytic in c and continuous in α. Notethat Φ(α, c) = 0 if and only if the system (4.1)–(4.2) has an unstable solution

φα(y) = φ1(y; α, c) + gr(c)φ2(y; α, c).

Since Φ(α0, c0) = 0 and the zeros of an analytic function are isolated, Φ(α0, c) 6=0 on the line |c − c0| = r2, where r2 > 0 is sufficiently small. By the continuityof Φ(α, c) in α, subsequently, Φ(α, c) 6= 0 for α ∈ Ir1 and |c − c0| = r2, wherer1, r2 > 0 are sufficiently small. Let us consider the function

N(α) =1

2πi

∮

|c−c0|=r2

∂Φ/∂c(α, c)Φ(α, c)

dc,

where α ∈ Ir1 . Observe that N(α) counts the number of zeros of Φ(α, c) in Br2 .Since N(α0) > 0 and N(α) is continuous as a function of α in Ir1 it follows thatN(α) > 0 for any α ∈ Ir1 . This proves that the set of unstable wave numbers isopen.

By definition, the boundary points of the set of unstable wave numbers must beneutral limiting wave numbers, say αs. Proposition 4.4 asserts that −α2

s must bea negative eigenvalue of the operator − d2

dy2 −K(y) on (0, h) with (4.5)–(4.6). Thiscompletes the proof. ¤


4.2. Proof of Theorem 4.2. The proof of Theorem 4.2 is to examine the bifurca-tion of unstable modes from each neutral limiting mode. This reduces to the studyof the bifurcation of zeros of the algebraic equation Φ(α, c) = 0 from (αs, cs) asis pointed out in the proof of Lemma 4.8. However, the differentiability of Φ(α, c)in c at a neutral limiting mode (αs, cs) can only be established in the directionof positive imaginary part, and thus the standard implicit function theorem doesnot apply. To overcome this difficulty, we construct a contraction mapping in sucha half neighborhood and prove the existence of an unstable mode near (αs, cs) asis done in [32] in the rigid-wall setting. Our proof, in addition, shows that theexistence of an unstable mode can only be established for wave numbers slightly tothe left of αs. Therefore, unstable modes to the left of αs continue to the zero wavenumber. Our method is similar to that in the rigid-wall setting in [32, Section 4].

Below we construct unstable solutions for wave numbers slightly to the left of aneutral limiting wave number.

Proposition 4.9. For U ∈ K+, let Us be the inflection value and y1, y2, . . . , yms

be the inflection points, as such U(yj) = Us for i = 1, 2, . . . , ms. If (φs, αs, Us) is aneutral limiting mode with αs positive, then for ε ∈ (ε0, 0), where |ε0| is sufficientlysmall, there exist φε and c(ε) such that the nontrivial function φε solves

(4.22) φ′′ε −(α2

s + ε)φε − U ′′

U − Us − c(ε)φε = 0 for y ∈ (0, h),

(4.23) φ′ε(h) = gr(Us + c(ε))φε(h) and φε(0) = 0

with Im c(ε) > 0. Moreover,

(4.24) limε→0−

c(ε) = 0,

(4.25)

limε→0−

dc

dε(ε) =

(∫ h

0

φ2sdy

)iπ

ms∑

j=1

K(yj)|U(yj)|φ

2s(yj) + p.v.

∫ h

0

K

U − Usφ2

sdy + A

−1

;

here p.v. means the Cauchy principal part and

(4.26) A =

2g

(U(h)−Us)3 + U ′(h)(U(h)−Us)2 if U(h) 6= Us

0 if U(h) = Us.

Proof. As in the proof of Lemma 4.8, for c ∈ C+ and ε < 0, let φ1(y; ε, c) andφ2(y; ε, c) be the solutions of

(4.27) φ′′ − (α2s + ε)φ− U ′′

U − Us − cφ = 0 for y ∈ (0, h)

normalized at h, that is

φ1(h) = 1, φ2(h) = 0,

φ′1(h) = 0, φ′2(h) = 1.

It is standard that φ1 and φ2 are analytic as a function of c in C+ and that φ1

and φ2 are linearly independent with Wronskian 1. The neutral limiting mode isnormalized so that φs(h) = 1 and φ′s(h) = gr(Us). The proof is again divided intotwo cases.


Case 1: U(h) 6= Us. Let us define

φ0(y; ε, c) = φ1(y; ε, c) + gr(Us + c)φ2(y; ε, c),

Φ(ε, c) = φ1(0; ε, c) + gr(Us + c)φ2(0; ε, c),

where gr is given in (4.7). It is readily seen that φ0 solves (4.27) with φ0(h) = 1and φ′0(h) = gr(Us + c). It is easy to see that Φ(ε, c) is analytic in c ∈ C+ anddifferentiable in ε. Note that an unstable solution to (4.22)–(4.23) exists if and onlyif Φ(ε, c) = 0 for some Im c > 0. The Green’s function of (4.27) is written as

G(y, y′; ε, c) = φ1(y; ε, c)φ2(y′; ε, c)− φ2(y; ε, c)φ1(y′; ε, c),

where φ1(y; ε, c) to be the solution of (4.27) with φ1(h) = 1 and φ′1(h) = gr(Us). Asimilar computation as in [32, pp. 336] for φj(y; ε, c) (j = 1, 2) yields that

(4.28)∂Φ∂ε

(ε, c) = −∫ h

0

G(y, 0; ε, c)φ0(y; ε, c)dy

and(4.29)

∂Φ∂c

(ε, c) =∫ h

0

G(y, 0; ε, c)−U ′′

(U − Us − c)2φ0(y; ε, c)dy +

d

dcgr(Us + c)φ2(0; ε, c).

Let us define the triangle in C+ as

∆(R,b) = cr + ici : |cr| < Rci , 0 < ci < band the Cartesian product in (0,∞)× C+ as

E(R,b1,b2) = (−b2, 0)×∆(R,b1),

where R, b1, b2 > 0 are to be determined later.We claim that:(a) For fixed R, both φ1(·; ε, c) and φ0(·; ε, c) uniformly converge to φs in C1[0, h]

as (ε, c) → (0, 0) in E(R,b1,b2). That is, for any δ > 0 there exists some b0 > 0 suchthat whenever b1, b2 < b0 and (ε, c) ∈ E(R,b1,b2) the inequalities

∥∥φ1(·; ε, c)− φs

∥∥C1 , ‖φ0(·; ε, c)− φs‖C1 6 δ

hold.(b) φ2(·; ε, c) converges uniformly in the sense in (a). Moreover, the limit function

φ2(y; 0, 0) has the boundary value φ2(0; 0, 0) = − 1φ′s(0) since the Wronskian of φ1

and φ2 and also the Wronskian of φs and φ2(·; 0, 0) are 1.Here, we prove the uniform convergence of φ0 only. The proof for φ1 and φ2

follows in the same way. Suppose on the contrary that there would exist a sequence(εk, ck)∞k=1 in E(R,b1,b2), where b1, b2 < b0 such that (εk, ck) → (0, 0) as k → ∞,yet

‖φ0(εk, ck)− φs‖C1 > δ0 > 0.

Since |Re ck| < R Im ck and Im ck < b0, it follows immediately that∥∥∥∥

U ′′

U − Us − ck

∥∥∥∥L∞

6 ‖K‖L∞

(1 +

∥∥∥∥ck

U − Us − ck

∥∥∥∥L∞

)

6 ‖K‖L∞(1 +√

R2 + 1)


and |gr(Us+ck)| 6 C independently of k. That is, φ0(·; εk, ck) satisfies the Rayleighequation (4.27), whose coefficients are uniformly bounded, and the boundary con-dition φ0(h) = 1, φ′0(h) = gr(Us + c), whose coefficients are uniformly bounded.Accordingly, ‖φ0(εk, ck)‖C2 is uniformly bounded, and by the Ascoli-Arzela theo-rem φ0(εk, ck) has a convergent subsequence in C1. That is, there exists φ∞ suchthat

‖φ0(εkj, ckj

)− φ∞‖C1 → 0 as kj →∞.

By continuity, φ∞ satisfies

φ′′∞ − α2sφ∞ − U ′′

U − Usφ∞ = 0 for y ∈ (0, h),

φ∞(h) = 1 and φ′∞(h) = gr(Us). This, however, implies φ∞ = φs and ‖φ0(εkj, ckj

)−φs‖C1 → 0. A contraction proves the claim.

In the appendix we prove that

(4.30)∂Φ∂ε

(ε, c) → 1φ′s(0)

∫ h

0

φ2sdy,

(4.31)∂Φ∂c

(ε, c) → − 1φ′s(0)

iπ

ms∑

j=1

K(yj)|U ′(yj)|φ

2s(yj) + p.v.

∫ h

0

K

U − Usφ2

sdy + A

uniformly as ε → 0− and c → 0 in E(R,b1,b2), where A is defined by (4.26) and yj

(j = 1, . . . , ms) are the inflection points for Us. Let us denote

B =1

φ′s(0)

∫ h

0

φ2sdy,

C = − 1φ′s(0)

(p.v.

∫ h

0

K(y)U − Us

φ2sdy + A

),(4.32)

D = − π

φ′s(0)

ms∑

j=1

K(yj)|U ′(yj)|φ

2s(yj).

Lemma 4.7 asserts that φs is nonzero at one of the inflection points, say φs(yj) 6= 0.Note that by [32, Remark 4.2], U ′(yj) 6= 0 for j = 1, 2, . . . , ms. Consequently,D < 0.

The remainder of the proof is identically the same as that of [32, Theorem 4.1]and hence we provide only its sketch. Let us define

f(ε, c) = Φ(ε, c)−Bε− (C + Di)c,(4.33)

F (ε, c) = − B

C + iDε− f(ε, c)

C + iD.(4.34)

Note that for each ε < 0 fixed a zero of an algebraic equation Φ(ε, ·) corresponds toa fixed point of the mapping F (ε, ·). Our goal is to show that for ε ∈ (ε0, 0), where|ε0| is sufficiently small, the mapping F (ε, ·) is contracting on ∆(R,b) for some Rand b.

On account of the uniform convergence of ∂Φ/∂ε and ∂Φ/∂c in (4.30) and (4.31),respectively, for any δ > 0 there exists b0 > 0 such that whenever b1, b2 < b0 theinequality

(4.35) |∂f/∂c|, |∂f/∂ε| < δ


holds in E(R,b1,b2). Subsequently, for any (ε, c), (ε′, c′) ∈ E(R,b1,b2), where b1, b2 <b0, it follows that

|f(ε, c)− f(ε′, c′)| 6 δ(|ε− ε′|+ |c− c′|).Since f(ε, c), Φ(ε, c) → 0 as (ε, c) → (0, 0), it follows that

(4.36) |f(ε, c)| 6 δ(|ε|+ |c|) for (ε, c) ∈ E(R,b1,b2).

Let us choose constants, as such in case C 6= 0,

R = 4|C/D| and δ =12

√C2 + D2 min

( |BC|Q(C2 + D2)

,−BD

Q(C2 + D2), 1

);

in case C = 0,

R = 1 and δ =12

√C2 + D2 min

( −BD

Q(C2 + D2), 1

);

Here, Q = −2BD√

R2 + 1(C2 + D2)−1 + 1. For such a δ > 0 choose b0 such that(4.35) and also (4.36) hold true whenever b1, b2 < b0. Let

b2 = b0 min(− C2 + D2

2BD√

R2 + 1, 1

), b1 = Qb2.

Finally, for ε ∈ (−b2, 0) fixed let

b(ε) = −2DB(C2 + D2)−1ε.

It is straightforward to show [32, pp. 339] that for each ε ∈ (−b2, 0) the mappingF (ε, ·) : ∆(R,b(ε)) → ∆(R,b(ε)) is contracting with a contraction ratio not greaterthan 1/2. That is, for each ε ∈ (−b2, 0) there exists a unique c(ε) ∈ ∆(R,b(ε))

such that F (ε, c(ε)) = c(ε). Since F (ε, ·) is analytic in ∆(R,b1) and contractinguniformly for ε, the fixed point c(ε) is in ∆(R,b1) and is differentiable in ε in theinterval (−b2, 0) (see, for instance, [13]). Let ε0 = −b2. Since c(ε) ∈ ∆(R,b(ε)) it isimmediate that (4.24) holds. Finally, differentiation of Φ(ε, c(ε)) = 0 yields

c′(ε) = −∂Φ/∂ε

∂Φ/∂c,

which, in view of (4.30) and (4.31), implies (4.25).Case 2: U(h) = Us. The proof is almost identical to that of Case 1, and thus we

only indicate some differences. Let us define

Φ(ε, c) = gs(Us + c)φ1(0; ε, c) + φ2(0; ε, c),

where gs is defined in (4.12). Note that φs(h) = φs(0) = 0. Thereby, the Green’sfunction is written as

G(y, y′; ε, c) = φ1(y; ε, c)φ2(y′; ε, c)− φ2(y; ε, c)φ1(y′; ε, c).

The same computations as in Case 1 yield that

∂Φ∂ε

→ − 1φ′s(0)

∫ h

0

φ2sdy

and

∂Φ∂c

(ε, c) → − 1φ′s(0)

iπ

ms∑

j=1

K(yj)|U ′(yj)|φ

2s(yj) + p.v.

∫ h

0

K

U − Usφ2

sdy

uniformly as ε → 0− and c → 0 in E(R,b1,b2). This completes the proof. ¤


Proof of Theorem 4.2. Let −α2N < −α2

N−1 < · · · < −α21 < 0 be negative eigen-

values of the operator − d2

dy2 −K(y) on (0, h) with boundary conditions (4.5)–(4.6.That is, αN = αmax, where −α2

max is defined either in (4.8a) or (4.8b). We de-duce from Lemma 4.8 and Proposition 4.9 that to each α ∈ (0, αN ) with α 6= αj

(j = 1, . . . , N) an unstable solution is associated. Our goal is to show the instabilityat α = αj for each j = 1, . . . , N − 1.

Case 1: U(h) 6= Us. Let (φk, αk, ck)∞k=1 be a sequence of unstable solutionssuch that αk → αj+ as k → ∞. After normalization, we may assume φk(h) = 1.Note that φk satisfies

(4.37) φ′′k − α2kφk − U ′′

U − ckφk = 0 for y ∈ (0, h)

and φ′k(h) = gr(ck), φk(0) = 0. Below we will prove that Im ck > δ > 0, where δ isindependent of k. Since the coefficients of (4.37) and gr(ck) are bounded uniformlyfor k, the solutions φk of the above Rayleigh equations are uniformly bounded inC2, and subsequently, φk converges in C2 as k → ∞, say to φ∞. The semicircletheorem (4.9) ensures that ck → c∞ as k → ∞. Note that Im c∞ > δ > 0. Bycontinuity, φ∞ satisfies

φ′′∞ − α2jφ∞ − U ′′

U − c∞φ∞ = 0 for y ∈ (0, h),

φ∞(h) = 1, φ′∞(h) = gr(c∞) and φ∞(0) = 0. That is, (φ∞, αj , c∞) is an unstablesolution of (4.1)–(4.2).

It remains to show that Im ck has a positive lower bound. Suppose on thecontrary that Im ck → 0 as k →∞.

We claim that ‖φk‖L2 6 C, where C > 0 is independent of k. Otherwise,‖φk‖L2 → ∞ as k → ∞. Let ϕk = φk/‖φk‖L2 so that ‖ϕk‖L2 = 1. Lemma4.6 then dictates that ‖ϕk‖H2 6 C independently of k. Subsequently, Proposition4.4 ensures that (ϕk, αk, ck) converges to a neutral limiting mode (ϕs, αs, Us). Bycontinuity, ‖ϕs‖L2 = 1 and

ϕ′′s − α2jϕs + K(y)ϕs = 0 for y ∈ (0, h).

On the other hand, ϕs(h) = ϕ′s(h) = 0 since ϕk(h) = 1/‖φk‖L2 → 0 and ϕ′k(h) =gr(ck)/‖φk‖L2 → 0 as k → ∞. Correspondingly, ϕs ≡ 0 on [0, h]. A contradictionproves the claim.

Since ‖φk‖L2 is bounded uniformly for k, Lemma 4.6 and Proposition 4.4 applyand ‖φk‖H2 6 C, ck → Us and φk → φs in C1, where φs satisfies

φ′′s − α2jφs − U ′′

U − Usφs = 0 for y ∈ (0, h)

with φs(h) = 1, φ′s(h) = gr(Us) and φs(0) = 0. An integration by parts yields that

0 =∫ h

0

(φs(φ′′k − α2

kφk − U ′′

U − ckφk)− φk(φ′′s − α2

jφs − U ′′

U − Usφs)

)dy

=(α2j − α2

k)∫ h

0

φsφk dy − (ck − Us)∫ h

0

U ′′

(U − ck)(U − Us)φsφkdy + gr(ck)− gr(Us).


Let us denote

Bk =∫ h

0

φsφkdy,

Dk = −∫ h

0

U ′′

(U − ck)(U − Us)φsφkdy +

gr(ck)− gr(Us)ck − Us

.

It is immediate that limk→∞Bk =∫ h

0|φs|2dy. We shall show in the appendix that

(4.38) limk→∞

Dk = A + iπ

ms∑

j=1

K(yj)|U(yj)|φ

2s(yj),

where A is defined by (4.26) and aj ’s (j = 1, 2, . . . , ms) are inflection points cor-responding to the inflection value Us. Since Im (limk→∞Dk) > 0 (see the proof ofProposition 4.9) it follows that

Im ck = (α2k − α2

j ) Im(Bk/Dk) < 0

for k large. A contradiction proves that Im ck has a positive lower bound, uni-formly for k.

Case 2: U(h) = Us. We normalize φk so that φ′k(h) = 1 and φk(h) = gs(ck).The proof is identically the same as that of Case 1 except that

Dk = −∫ h

0

U ′′

(U − ck)(U − Us)φsφk dy +

gs(ck)ck − Us

.

We shall show in the appendix that

(4.39) limk→∞

Dk = iπ

ms∑

j=1

K(yj)|U ′(yj)|φ

2s(yj) + p.v.

∫ h

0

K

U − Usφ2

sdy.

This proves that there exists an unstable solution for each α ∈ (0, αmax).It remains to prove linear stability in case either the operator − d2

dy2 −K(y) ony ∈ (0, h) with (4.5)–(4.6) is nonnegative or α > αmax. Suppose otherwise, thereexists an unstable mode at a wave number α > αmax. By Lemma 4.8, we cancontinuate this unstable mode for wave numbers larger than α until the growthrate becomes zero. By Lemma 6.3, this continuation must stop at a wave numberαs > α, where there is a neutral limiting mode. Then by Proposition 4.4, −α2

s is anegative eigenvalue of − d2

dy2 −K (y) with (4.5)-(4.6). But −α2s < −α2

max which is a

contradiction to the fact that −α2max is the lowest eigenvalue of − d2

dy2 −K (y) with(4.5)-(4.6).

This completes the proof. ¤

Remark 4.10. For U ∈ K+, let −α2d be the lowest eigenvalue of − d2

dy2 − K(y) ony ∈ (0, h) with the Dirichlet boundary conditions φ(h) = 0 = φ(0). If U(h) = Us

then αmax = αd. We claim that if U(h) 6= Us then αmax > αd. To see this, letφd be the eigenfunction of − d2

dy2 −K(y) on y ∈ (0, h) with the Dirichlet boundary

conditions corresponding to −α2d and let φm be the eigenfunction of − d2

dy2 −K(y)with the boundary conditions (4.5)–(4.6) corresponding to −α2

max. By Sturm’s


theory, we can assume φd, φm > 0 on y ∈ (0, h). An integration by parts yields that

0 =∫ h

0

(φd(φ′′m − α2

maxφm + K(y)φm)− φm(φ′′d − α2dφd + K(y)φd)

)dy

= −φm(h)φ′d(h) + (α2d − α2

max)∫ h

0

φdφmdy.

Since φ′d(h) < 0 the claim follows.For flows in class K+, the lowest eigenvalue of (4.4) measures the range of in-

stability, for both the free surface and the rigid wall cases. That means, (0, αmax)is the interval of unstable wave numbers in the free-surface setting (Theorem 4.2)and (0, αd) is the interval of unstable wave numbers in the rigid-wall setting [32,Theorem 1.2]. The fact that αmax > αd thus indicates that the free surface has adestabilizing effect.

4.3. Monotone unstable shear flows. In general, for a given shear flow profilewith one inflection value, one show the existence of a neutral limiting mode, whichis contiguous to exponentially growing modes, and thus the linear instability bynumerically computing the negativity of (4.8). For a monotone flow, however, acomparison argument establishes a general statement.

Lemma 4.11. For any monotone shear flow with exactly one inflection point ys

in the interior, there exists a neutral limiting mode. That is, (4.1)–(4.2) has anontrivial solution for which c = U(ys) and α > 0.

Proof. This result is given in Theorem 4 of [48]. Here we present a detailed prooffor completeness and also for clarification of some arguments in [48].

W consider an increasing flow U(y) in y only. A decreasing flow in y can betreated in the same way. Let Us = U(ys) be the inflection value. Denoted by φα isthe solution of the Rayleigh equation

φ′′α +(

U ′′

Us − U− α2

)φα = 0 for y ∈ (0, h)

with φα(0) = 0 and φ′α(0) = 1. As in the proof of Lemma 2.5, integrating the aboveover (0, h) yields that

(U(h)− Us)φ′α(h)− (U(0)− Us)− φα(h)U ′(h)− α2

∫ h

0

(U − Us)φαdy = 0,

and thusφ′α(h)φα(h)

=U(0)− Us

(U(h)− Us)φα(h)+

U ′(h)U(h)− Us

+α2

(U(h)− Us)φα(h)

∫ h

0

(U − Us)φαdy.

It is straightforward to see that the boundary condition (4.2) is satisfies if and onlyif the function

f(α) =U(0)− Us

(U(h)− Us)φα(h)+

α2

(U(h)− Us)φα(h)

∫ h

0

(U − Us)φαdy − g

(Us − U(h))2.

vanishes at some α > 0.We claim that φα(y) > 0 on y ∈ (0, h] for any α > 0. Suppose on the contrary

that yα ∈ (0, h] to be the first zero of φα other than 0, that is, φα(yα) = 0 andφα(y) > 0 for y ∈ (0, yα). Then, yα > ys must hold. Indeed, if yα 6 ys were tobe true, then Sturm’s first comparison theorem would apply to φα and U − Us on


[0, yα] to assert that U −Us would vanish somewhere in (0, yα) ⊂ (0, ys). This usesthat

(U − Us)′′ +U ′′

Us − U(U − Us) = 0.

A contradiction then asserts that yα > ys. Correspondingly, φα and U − Us haveexactly one zero in [0, yα]. On the other hand, by Sturm’s second comparisontheorem [25], it follows that

φ′α(yα)φα(yα)

> U ′(yα)U(yα)− Us

.

This contradicts since φα(yα) = 0 and the left hand side is −∞. Therefore, φα(y) >0 for y ∈ (0, h] and for any α > 0. In particular, φα(h) > 0 for any α > 0.Consequently, f is a continuous function of α and f(0) < 0.

It remains to show that f(α) > 0 for α > 0 big enough. Thereby, by continuityf vanishes at some α > 0. Let

∣∣∣ U ′′Us−U

∣∣∣ 6 M and α2 > M . Le us denote by φ1 andφ2 the solutions of

φ′′1 + (M − α2)φ1 = 0 and φ′′2 + (−M − α2)φ2 = 0 for y ∈ (0, h),

respectively, with φi(0) = 0 and φ′i(0) = 1. It is straightforward that

φ1(y) =1√

α2 −Msinh

√α2 −My and φ2(y) =

1√α2 + M

sinh√

α2 + My.

As in the proof of Lemma 2.7 (a), Sturm’s second comparison theorem [25] impliesthat

1√α2 −M

sinh√

α2 −My 6 φα(y) 6 1√α2 + M

sinh√

α2 + My.

This together with the monotone property of U establishes that f(α) > C1α− C2

for some constants C1, C2 > 0. For details we refer to [48, Theorem 4]. Thus,f(α) > 0 if α > 0 is sufficiently large. This completes the proof . ¤

Since a monotone flow with one inflection value is in class K+, the above lemmacombined with Theorem 4.2 asserts its instability.

Corollary 4.12. Any monotone shear flow with exactly one inflection point inthe interior is unstable in the free-surface setting for wave numbers in an interval(0, αmax) with αmax > 0.

Remark 4.13. In the free-surface setting, there are two different kinds of neutralmodes, solutions to the Rayleigh system (4.1)–(4.2) with Im c = 0. Neutral limitingmodes have their phase speed in the range of the shear profile. In class K+, more-over, the phase speed of a neutral limiting mode must be the inflection value of theshear profile (Proposition 4.4) and it is contiguous to unstable modes (Proposition4.9). On the other hand, Lemma 2.5 shows that under the conditions U ′′(h) < 0and U(h) > U(y) for h 6= y a neutral mode exists with the phase speed c > max U .Such a neutral mode is used in the local bifurcation of nontrivial periodic waves inTheorems 2.2 and 2.6. In view of the semicircle theorem, however, such a neutralmode is not contiguous to unstable modes. This means, neutral modes governingthe stability property are different from those governing the bifurcation of non-trivial waves. This is an important difference in the free-surface setting. Since inthe rigid wall setting, for any possible neutral modes the phase speed must lie in


the range of U , which follows easily from Sturm’s first comparison theorem. Forclass K+ flows, such neutral modes are contiguous to unstable modes ([32]) and thebifurcation of nontrivial waves from these neutral modes can also be shown ([7]).Thus, in the rigid-wall setting, the same neutral modes govern both stability andbifurcation.

Remark 4.14. In [15], the J -formal stability was introduced via a quadratic form,which is related to the local bifurcation of nontrivial waves in the transformedvariables (see also Introduction), and it was concluded that this formal stability ofthe trivial solutions switches exactly at the bifurcation point. Below we discuss twoexamples for which the linear stability property does not change along the line oftrivial solutions passing the bifurcation point, which indicates that the J -formalstability in [15] is unrelated to linear stability of the physical water wave problem.However, the J -formal stability results [15] does give more information about thestructure of the periodic water wave branch.

Let us consider a monotone increasing flow U(y) on y ∈ [0, h] with one inflectionpoint ys ∈ (0, h), for example, U(y) = a sin b(y − h/2) on y ∈ [0, h] for whichys = h/2. By lemma 4.11 and Corollary 4.12, such a shear flow is equipped with aneutral limiting mode with c = U(ys) and α = αmax > 0, and it is linearly unstablefor any wave number α ∈ (0, αmax). In addition, by lemma 2.5 and Theorem2.6 for any wave number α ∈ (0, αmax) this shear flow has a neutral mode withc(α) > U(h) = max U . Moreover, such a neutral mode is a nontrivial solution to thebifurcation equation (2.9)–(2.10), and thus there exists a local curve of bifurcationof nontrivial waves with a wave speed c(α) and period 2π/α. Let p0 and γ be theflux and vorticity relation determined by U (y) , c(α) and h via (2.11). Consider thetrivial solutions with shear flows U(y; µ) defined in Lemma 2.1, with above p0, γand the parameter µ. The bifurcation point U(y; µ0) = U (y)− c(α) corresponds toµ0 = (U(h)− c(α))2. The instability of U(y; µ0) at the wave number α continuatesto shear flows U(y; µ) with µ near µ0, which can be shown by a similar argumentas in the proof of Lemma 4.8. So at the bifurcation point µ0, there is NO switchof stability of trivial solutions.

Let us consider U ∈ C2([0, h]) satisfying that U ′(y) > 0 and U ′′(y) < 0 iny ∈ [0, h]. For such a shear flow Lemma 2.5 and Theorem 2.6 applies as well andthere exists a local curve of bifurcation of nontrivial waves for any wave numberα which travel at the speed c(α) > maxU , where c(α) is chosen so that the bifur-cation equation (2.9)–(2.10) is solvable. For this bifurcation flow U (y)− c(α), thevorticity relation γ determined via (2.11) is monotone since U ′′ does not changesign. Consequently, any shear flows U(y; µ) defined in Lemma 2.1 with the sameγ has no inflection points, as also remarked at the end of Section 3. Therefore,by Theorem 6.4 all trivial solutions corresponding to these shear flows U(y; µ) arestable. This again shows that the bifurcation of nontrivial periodic waves does notinvolve the switch of stability of trivial solutions.

5. Linear instability of periodic water waves with free surface

We now turn to investigating the linear instability of periodic traveling wavesnear an unstable background shear flow. Suppose that a shear flow (U(y), 0) withU ∈ C2+β([0, h0]) and U ∈ K+ has an unstable wave number α > 0, that is,for such a wave number α > 0 the Rayleigh system (4.1)–(4.2) has a nontrivialsolution with Im c > 0. Suppose moreover that for the unstable wave number


α > 0 the bifurcation equation (2.9)–(2.10) is solvable with some c(α) > max U .Then Remark 2.4 and Theorem 2.2 apply to state that there exists a one-parametercurve of small-amplitude traveling-wave solutions (ηε(x), ψε(x, y)) satisfying (2.2)of period 2π/α and the wave speed c(α), where ε > 0 is the amplitude parameter. Anatural question is: are these small-amplitude nontrivial periodic waves generatedover the unstable shear flow also unstable? The answer is YES under some technicalassumptions, which is the subject of the forthcoming investigation.

5.1. The main theorem and examples. We prove the linear instability of thesteady periodic water-waves (ηε(x), ψε(x, y)) by finding a growing-mode solution tothe linearized water-wave problem. As in Section 3, let

Dε = (x, y) : 0 < x < 2π/α, 0 < y < ηε(x) and Sε = (x, ηε(x)) : 0 < x < 2π/αdenote, respectively, the fluid domain of the steady wave (ηε(x), ψε(x, y)) of oneperiod and the steady surface. The growing-mode problem (3.8) of the linearizedperiodic water-wave problem around (ηε(x), ψε(x, y)) reduces to

∆ψ + γ′(ψε)ψ − γ′(ψε)∫ 0

−∞λeλsψ(Xε(s), Yε(s))ds = 0 in Dε;(5.1a)

λη(x) +d

dx

(ψεy(x, ηε(x))η(x)

)= − d

dxψ(x, ηε(x));(5.1b)

P (x, ηε(x)) + Pεy(x)η(x) = 0;(5.1c)

λψn(x) +d

dx

(ψεy(x, ηε(x))ψn(x)

)= − d

dxP (x, ηε(x))− Ω

d

dxψ(x, ηε(x));(5.1d)

ψ(x, 0) = 0.(5.1e)

Here and in sequel, let us abuse notation and denote that Pεy(x) = Pεy(x, ηε(x)),that is, Pεy(x, y) restricted on the steady wave-profile y = ηε(x). By Theorem 2.2,Pεy(x) = Pεy(x, ηε(x)) = −g + O(ε). Recall that

ψn(x) = ∂yψ(x, ηε(x))− ηεx(x)∂xψ(x, ηε(x))

is the derivative of ψ(x, ηε(x)) in the direction normal to the free surface (x, ηε(x))and that Ω = γ(0) is the vorticity of the steady flow of ψε(x, y) on the steadywave-profile y = ηε(x). Note that Ω is a constant independent of ε.

Theorem 5.1 (Linear instability of small-amplitude periodic water-waves). LetU ∈ C2+β([0, h0]), β ∈ (0, 1), be in class K+. Suppose that U(h0) 6= Us, whereUs is the inflection value of U , and that αmax defined by (4.8a) is positive, assuch Theorem 4.2 applies to find the interval of unstable wave numbers (0, αmax).Suppose moreover that for some α ∈ (αmax/2, αmax) there exists c(α) > max U suchthat the bifurcation equation (2.9)– (2.10) has a nontrivial solution. Let us denoteby (ηε(x), ψε(x, y)) the family of nontrivial waves with the period 2π/α and the wavespeed c(α) bifurcating from the trivial solution η0(x) ≡ h0 and (ψ0y(y),−ψ0x(y)) =U(y)− c(α), 0), where ε > 0 is the small-amplitude parameter. Provided that

(5.2) g + U ′(h0)(U(h0)− Us) > 0,

then for each ε > 0 sufficiently small, there exists an exponentially growing solu-tion (eλtη(x), eλtψ(x, y)) of the linearized system (3.5), where Re λ > 0, with theregularity property

(η(x), ψ(x, y)) ∈ C2+β([0, 2π/α])× C2+β(Dε).


Remark 5.2 (Examples). As is discussed in Remark 4.14, any increasing flow shearflow U ∈ C2+β([0, h0]), β ∈ (0, 1), with exactly one inflection point in y ∈ (0, h0)satisfies αmax > 0. Moreover, Lemma 2.5 applies and small-amplitude periodicwaves bifurcate at any wave number α > 0. Since (5.2) holds true, therefore, byTheorem 5.1 small-amplitude periodic waves bifurcating from such a shear flow atany wave number α ∈ (αmax/2, αmax) are unstable. An example includes

(5.3) U(y) = a sin b(y − h0/2) for y ∈ [0, h0],

where h0, b > 0 satisfy h0b 6 π and a > 0 is arbitrary.(1) The shear flow in (5.3) is unstable under periodic perturbations of a wave

number α ∈ (0, αmax), where αmax > 0. Note that in the rigid-wall setting [32], thesame shear flow is stable under perturbations of any wave number. This indicatesthat free surface has a destabilizing effect. This serves as an example of Remark4.10 since αmax > 0 and αd = 0.

(2) The amplitude a and the depth h0 in (5.3) may be chosen arbitrarily small,and the shear flow as well as the nontrivial periodic waves near the shear flow areunstable for any wave number α ∈ (0, αmax), which conflicts with the result in [15]that small-amplitude rotational periodic water-waves are J -formally stable if thevorticity strength and the depth are sufficiently small. Thus, as also commentedin Remark (4.14), the J -formal stability in [15] is not directly related to the linearstability of water waves. Indeed, while ∂J (η, ψ) = 0 gives the equations for steadysteady waves, the linearized water-wave problem is not in the form

∂t(η, ψ) = (∂2J )(η, ψ),

which is implicitly required in [15] in order to apply the Crandall-Rabanowitz theory[16] of exchange of stability.

(3) Our example (5.3) also indicates that adding an arbitrarily small vorticityto the irrotational water wave system of an arbitrary depth may induce instability.That means, although small irrotational periodic waves are found to be stableunder perturbations of the same period [39], [45], they are not structurally stable;Vorticity has a subtle influence on the stability of water waves.

The proof of Theorem 5.1 uses a perturbation argument. At ε = 0 the trivialsolution (η0(x), ψ0(x, y)) corresponds to the shear flow (U(y) − c(α), 0) under theflat surface y = h0. For the simplicity of notations, in the remainder of thissection, we write U(y) for U(y)− c(α), as is done in Section 4. Thereby, U(y) < 0.Since α is an unstable wave number of U(y), there exist an unstable solution φα tothe Rayleigh system (4.1)–(4.2) and an unstable phase speed cα. That is, φα 6≡ 0,Im cα > 0 and

φ′′α − α2φα +U ′′

U − cαφα = 0 for y ∈ (0, h0),

φ′α(h0) =(

g

(U(h0)− cα)2+

U ′(h0)U(h0)− cα

)φα(h0),

φα(0) = 0.

(5.4)

This corresponds to a growing mode solution satisfying (5.1) at ε = 0, whereλ0 = −iαcα has a positive real part. Our goal is to show that for ε > 0 suffi-ciently small, there is found λε near λ0 such that the growing-mode problem (5.1)


at (ηε(x), ψε(x, y)) is solvable. First, the system (5.1) is reduced to an opera-tor equation defined in a function space independent of ε. Then, by showing thecontinuity of this operator with respect to the small-amplitude parameter ε, thecontinuation of the unstable mode follows from the eigenvalue perturbation theoryof operators.

5.2. Reduction to an operator equation. The purpose of this subsection is toreduce the growing mode system (5.1) to an operator equation on L2

per(Sε). Hereand elsewhere the subscript per denotes the periodicity in the x-variable. The ideais to express η(x) on Sε and ψ(x, y) in Dε (and hence P (x, ηε(x))) in terms ofψ(x, ηε(x)).

Our first task is to relate η(x) with ψ(x, ηε(x)).

Lemma 5.3. For |λ−λ0| 6 (Re λ0)/2, where λ0 = −iαcα, let us define the operatorCλ : L2

per(Sε) → L2per(Sε) by

Cλφ(x) = − 1ψεy(x)

φ(x) +1

ψεy(x)eλa(x)

∫ x

0

λeλa(x′)ψ−1εy (x′)φ(x′)dx′

− λ

ψεy(x)eλa(x)(1− eλa(2π/α)

)∫ 2π/α

0

eλa(x′)ψ−1εy (x′)φ(x′)dx′,

(5.5)

where a(x) =∫ x

0ψ−1

εy (x′, ηε(x′))dx′. For simiplicity, here and in the sequel weidentify ψey(x) with ψey(x, ηε(x)) and φ(x) with φ(x, ηε(x)), etc. Then,

(a) The operator Cλ is analytic in λ for |λ− λ0| 6 (Re λ0)/2, and the estimate∥∥Cλ

∥∥L2

per(Sε)→L2per(Sε)

6 K

holds, where K > 0 is independent of λ and ε.(b) For any φ ∈ L2

per(Sε), the function ϕ = Cλφ is the unique L2per(Sε)- weak

solution of the first-order ordinary differential equation

(5.6) λϕ +d

dx(ψεy(x)ϕ) = − d

dxφ.

If, in addition, φ ∈ C1per(Sε) then ϕ ∈ C1

per(Sε) is the unique classical solution of(5.6).

Proof. Assertions of (a) follows immediately since ψεy(x) < 0 and thus a(x) < 0and since Re λ > Re λ0.

(b) First, we consider the case φ ∈ C1per(Sε) to motivate the definition of Cλ.

Let us write (5.6) as the first-order ordinary differential equation

d

dxϕ +

1ψεy

(λ +

d

dxψεy

)ϕ = − 1

ψεy

d

dxφ,

which has a unique 2π/α-periodic solution

ϕ(x) = − 1ψεy(x)eλa(x)

( ∫ x

0

eλa(x′) d

dxφ(x′)dx′

− 11− eλa(2π/α)

∫ 2π/α

0

eλa(x′) d

dxφ(x′)dx′

).

An integration by parts of the above formula yields that ϕ(x) = Cλφ is as definedin (5.5).


In case φ ∈ L2per(Sε), the integral representation (5.5) makes sense and solves

(5.6) in the weak sense. Indeed, an integration by parts shows that ϕ defined by(5.5) satisfies the weak form of equation (5.6)

∫ 2π/α

0

(λϕ(x)h(x)− ψεy(x)ϕ(x)

d

dxh(x)− φ(x)

d

dxh(x)

)dx = 0

for any 2π/α-periodic function h ∈ H1per([0, 2π/α]). In order to show the unique-

ness, suppose that ϕ ∈ L2per(Sε) is another weak solution of (5.6). Let ϕ1 = ϕ− ϕ.

Then, ϕ1 ∈ L2per(Sε) is a weak solution of the homogeneous differential equation

λϕ1 +d

dx(ψεy(x)ϕ1) = 0.

It is readily seen that∫ 2π/α

0ϕ1(x)dx = 0. Note that h1(x) =

∫ x

0ϕ1(x′)dx′ defines

2π/α-periodic function in H1per([0, 2π/α]). Multiplication of the above homogeneous

equation by (λh1)∗ and an integration by parts then yields that

0 = Re∫ 2π/α

0

(|λ|2 ϕ1(x)h∗1(x)− ψεy(x)λ∗ϕ1(x)

(d

dxh1(x)

)∗)dx

= |λ|2∫ 2π/α

0

d

dx

(12 |h1|2

)dx− Re λ

∫ 2π/α

0

ψεy(x) |ϕ1|2 dx

= −Re λ

∫ 2π/α

0

ψεy(x) |ϕ1|2 dx.

Here and elsewhere, the asterisk denotes the complex conjugation. Since Reλ > 0and ψεy(x) < 0, it follows that ϕ1 ≡ 0, and in turn, ϕ ≡ ϕ. This completes theproof. ¤

Note that the range of Cλ is of mean-zero. That is,∮Sε

(Cλφ)(x)dx = 0. Indeed,integrating (5.6) with ϕ = Cλφ proves the assertion. Formally, the operator Cλ iswritten as

Cλφ(x) = −(

λ +d

dx(ψεy(x)φ(x))

)−1d

dxφ(x).

Let us denote f(x) = ψ(x, ηε(x)). With the use of Cλ then the boundary condi-tions (5.1b)–(5.1d) are written in terms of f as

η(x) = Cλf(x),

P (x, ηε(x)) = −Pεy(x)Cλf(x),

ψn(x) = −Cλ(Pεy(x)Cλ + Ω id)f(x),

where id : L2(Sε) → L2(Sε) is the identity operator.Our next task is to relate ψ(x, y) in Dε with f(x) = ψ(x, ηε(x)). Given b ∈

L2per(Sε) let ψb ∈ H1(Dε) be a weak solution of the elliptic partial differential

equation

∆ψ + γ′(ψε)ψ − γ′(ψε)∫ 0

−∞λeλsψ(Xε(s), Yε(s))ds = 0 in Dε(5.7a)

ψn(x) := ∂yψ(x, ηε(x))− ηεx(x)∂xψ(x, ηε(x)) = b(x) on Sε,(5.7b)

ψ(x, 0) = 0(5.7c)


such that ψb is 2π/α-periodic in the x-variable. Lemma 5.6 below proves that theboundary value problem (5.7) is uniquely solvable and ψb ∈ H1(Dε) provided that|λ − λ0| 6 (Re λ0)/2 and (ηε(x), ψε(x, y)) is near the trivial solution with the flatsurface y = h0 and the unstable shear flow (U(y), 0) given in Theorem 5.1. This,together with the trace theorem, allows us to define an operator Tε : L2

per(Sε) →L2

per(Sε) by

(5.8) Tεb(x) = ψb(x, ηε(x)),

which is the unique solution ψb of (5.7) restricted on the steady surface Sε.Be definition, it follows that

f(x) = ψ(x, ηε(x)) = Tεψn(x).

This, together with the boundary conditions written in terms of f as above yieldsthat

(5.9) f = −TεCλ(Pεy(x)Cλ + Ω id)f.

The growing-mode problem (5.1) is thus reduced to find a nontrivial solution f(x) =ψ(x, ηε(x)) ∈ L2

per(Sε) of the equation (5.9), or equivalently, to show that theoperator

id + TεCλ(Pεy(x)Cλ + Ω id)has a nontrivial kernel for some λ ∈ C with Re λ > 0.

The remainder of this subsection concerns with the unique solvability of (5.7).Our first task is to compare (5.7) at ε = 0 with the Rayleigh system, which is usefulin later consideration.

Lemma 5.4. For U ∈ K+ for y ∈ [0, h0] and U(h0) 6= Us, let −α2max be the lowest

eigenvalue of − d2

dy2 −K(y) for y ∈ (0, h0) subject to the boundary conditions

(5.10) φ′(h0) =(

g

(U(h0)− Us)2+

U ′(h0)U(h0)− Us

)φ(h0) and φ(0) = 0,

where K is defined in (4.3). Let −α2n be the lowest eigenvalue of − d2

dy2 −K(y) fory ∈ (0, h0) subject to the boundary conditions

(5.11) φ′(h0) = 0 and φ(0) = 0.

If g + U ′(h0)(U(h0)− Us) > 0, then αmax > αn.

Proof. The argument is nearly identical to that in Remark 4.10. Let us denote byφm the eigenfunction of − d2

dy2 −K(y) on y ∈ (0, h0) with (5.10) corresponding to

the eigenvalue −α2max and by φn the eigenfunction of − d2

dy2 −K(y) on y ∈ (0, h0)with (5.11) corresponding to the eigenvalue −α2

n. By the standard theory of Sturm-Liouville systems we may assume φm > 0 and φn > 0 on y ∈ (0, h0). An integrationby parts yields that

0 =∫ h0

0

(φn(φ′′m − α2

maxφm + K(y)φm)− φm(φ′′n − α2nφn + K(y)φn)

)dy

=(α2n − α2

max)∫ h0

0

φnφmdy +g + U ′(h0)(U(h0)− Us)

(U(h0)− Us)2φm(h0)φn(h0).

The assumption g + U ′(h0)(U(h0)− Us) > 0 then proves the assertion. ¤Our next task is the unique solvability of the homogeneous problem of (5.7).


Lemma 5.5. Assume that U(h0) 6= Us and g +U ′(h0)(U(h0)−Us) > 0. For ε > 0sufficiently small and |λ− λ0| 6 (Re λ0)/2, the following elliptic partial differentialequation

(5.12a) ∆ψ + γ′(ψε)ψ − γ′(ψε)∫ 0

−∞λeλsψ(Xε(s), Yε(s))ds = 0 in Dε

subject to

(5.12b) ψ(x, 0) = 0

and the Neumann boundary condition

(5.12c) ψn = 0 on Sε

admits only the trivial solution ψ ≡ 0.

A main difficulty in the proof of Lemma 5.5 is that the domain Dε depends onthe small amplitude parameter ε > 0 whereas the statement of Lemma 5.5 calls foran estimate of solutions of the system (5.12) uniform for ε > 0. In order to compare(5.12) for different values of ε, we employ the action-angle variables, which map thedomain Dε into a common domain independent of ε. For any (x, y) ∈ Dε, let usdenote by (x′, y′) : ψε(x′, y′) = ψε(x, y) = p the streamline containing (x, y), byσ the arc-length variable on the streamline ψε(x′, y′) = p, and by σ(x, y) thevalue of σ corresponding to the point (x, y) along the streamline. Let us define thenormalized action-angle variables as(5.13)

I =α

2π

h0

hε

∫∫

ψε(x′,y′)<pdy′dx′, and θ = υε(I)

∫ σ(x,y)

0

1|∇ψε|

∣∣∣ψε(x′y′)=p

dσ′,

where h0 and hε are the mean water depth at the parameter values 0 and ε, respec-tively, and

υε(I) =2π

α

(∮

ψε(x′,y′)=p

1|∇ψε|

)−1

.

The action variable I represents the (normalized) area in the phase space under thestreamline (x′, y′) : ψε(x′, y′) = ψε(x, y) = p and the angle variable θ representsthe position along the streamline of ψε(x, y). The assumption of no stagnation,i.e. ψεy(x, y) < 0 throughout Dε, implies that all stream lines are non-closed. For,otherwise, the horizontal velocity ψεy must change signs on a closed streamline.Moreover, for ε > 0 sufficiently small, all streamlines are close to those of the trivialflow, that is, they almost horizontal. Therefore, the action-angle variable (θ, I) isdefined globally in Dε. The mean-zero property (2.8) of the wave profile ηε(x)implies that the area of the (steady) fluid region Dε is (2π/α)hε. Accordingly, bythe definition in (5.13) it follows that 0 < I < h0 and 0 < θ < 2π/α, independentlyof ε.

Let us define the mapping by the action-angle variables by Aε(x, y) = (θ, I).From the above discussions follows that Aε is bijective and maps Dε to

D = (θ, I) : 0 < θ < 2π/α, 0 < I < h0.At ε = 0, the action-angle mapping reduces to the identity mapping on D0 = D.For ε > 0, the mapping has a scaling effect. More precisely,∫∫

D

f(A−1ε (θ, I))dθdI =

h0

hε

∫∫

Dε

f(x, y)dydx


for any function f defined in Dε.Another motivation to employ the action-angle variables comes from that they

simplify the equation on the particle trajectory. In the action-angle variables (θ, I),the characteristic equation (3.7) becomes [3, Section 50]

θ = −υε(I)I = 0,

where the dot above a variable denotes the differentiation in the σ-variable. Thisobservation will be useful in future considerations.

Proof of Lemma 5.5. Suppose on the contrary that there would exist sequencesεk → 0+, λk → λ0 as k → ∞ and ψk ∈ H1(Dεk

) such that ψk 6≡ 0 is a solution of(5.12) with ε = εk. After normalization, ‖ψk‖L2(Dεk

) = 1. We claim that

(5.14) ‖ψk‖H2(Dεk) 6 C,

where C > 0 is independent of k. Indeed, by Minkovski’s inequality it follows that∥∥∥∥γ′(ψεk

)ψk − γ′(ψεk)∫ 0

−∞λeλsψk(Xεk

(s), Yεk(s))ds

∥∥∥∥L2(Dεk

)

6 ‖γ′(ψεk)‖L∞

(‖ψk‖L2(Dεk

) +∫ 0

−∞|λ| eRe λs ‖ψk(Xεk

(s), Yεk(s))‖L2(Dεk

) ds

)(5.15)

= ‖γ′(ψεk)‖L∞‖ψk‖L2

(1 +

|λ|Re λ

)6 ‖γ′(ψεk

)‖L∞

(1 +

|λ0|+ δ

δ

).

This uses the fact that the mapping (x, y) → (Xεk(s), Yεk

(s)) is measure preservingand that |λ − λ0| < Re λ0/2. The standard elliptic regularity theory [2] for aNeumann problem adapted for (5.12) then proves the estimate (5.14). This provesthe claim.

In order to study the convergence of ψk, we perform the mapping by theaction-angle variables (5.13) and write Aεk

(x, y) = (θ, I). Note that the image ofDεk

under the mapping Aεkis

D = (θ, I) : 0 < θ < 2π/α , 0 < I < h0,which is independent of k. Let us denote Aψk(θ, I) = ψk(A−1

εk(θ, I)). It is immedi-

ate to see that Aψk ∈ H2(D).Since hε = h0 + O(ε) it follows that

‖Aψk‖L2(D) = (h0/hε)‖ψk‖L2(Dεk) = 1 + O(ε).

This, together with (5.14), implies that

Aψk → ψ∞ weakly in H2(D) as k →∞,

Aψk → ψ∞ strongly in L2(D) as k →∞for some ψ∞. By continuity, ‖ψ∞‖L2(D0) = 1. Our goal is to show that ψ∞ ≡ 0and thus prove the assertion by contradiction.


At the limit as εk → 0, the limiting mapping A0 is the identity mapping onD = D0 and the limit function ψ∞ satisfies

∆ψ∞ + γ′(ψ0)ψ∞ − γ′(ψ0)∫ 0

−∞λ0e

λ0sψ∞(X0(s), Y0(s))ds = 0 in D;

∂yψ∞(x, h0) = 0;(5.16)

ψ∞(x, 0) = 0.

Here, λ0 = −iαcα and (X0(s), Y0(s)) = (x + U(y)s, y).Since the above equation and the boundary conditions are separable in the x

and y variables, ψ∞ can be written as

ψ∞(x, y) =∞∑

l=0

eilαxφl(y).

In case l = 0, the boundary value problem (5.16) reduces to

φ′′0 = 0 for y ∈ (0, h0)φ′0(h0) = 0, φ0(0) = 0,

and thus, φ0 ≡ 0.Next, consider the solution φ1 of (5.16) when l = 1:

φ′′1 − α2φ1 − U ′′

U−cαφ1 = 0 for y ∈ (0, h0)

φ′1(h0) = 0, φ1(0) = 0.

This uses that γ′(ψ0) = −U ′′/U . Recall that to the unstable wave number α andthe unstable wave speed cα is associated an unstable solution φα satisfying (5.4).As is done in the proof of Lemma 5.4, an integration by parts yields that

0 =∫ h0

0

(φ1

(φ′′α − α2φα − U ′′

U − cαφα

)− φα

(φ′′1 − α2φ1 − U ′′

U − cαφ1

))dy

=(

g

(U(h0)− cα)2+

U ′(h0)U(h0)− cα

)φα(h0)φ1(h0).

Since φα(h0) 6= 0, this implies φ1(h0) = 0 and, in turn, φ1 ≡ 0. In addition,φ′1(h0) = 0.

For l > 2, the solution φl of (5.16) ought to satisfy

φ′′l − l2α2φl − U ′′U−cα/lφl = 0 for y ∈ (0, h0)

φ′l(h0) = 0, φl(0) = 0.

An integration by parts yields that∫ h0

0

(|φ′l|2 + l2α2|φl|2 +

U ′′

U − cα/l|φl|2

)dy = 0,

and subsequently, for any q real follows that (see the proof of Lemma 4.6)∫ h0

0

(|φ′l|2 + l2α2|φl|2 +

U ′′(U − q)|U − cα/l|2 |φl|2

)dy = 0.

The same argument as in Lemma 4.6 applies to assert that∫ h0

0

(|φ′l|2 + l2α2|φl|2)dy 6∫ h0

0

K(y)|φl|2dy.


Recall that K(y) = −U ′′(y)/(U(y)− Us) > 0.Let −α2

n be as in Lemma 5.4 the lowest eigenvalue of d2

dy2 −K(y) on y ∈ (0, h0)with the boundary conditions φ′(h0) = 0 and φ(0) = 0. By Lemma 5.4, αmax > αn.On the other hand, the variational characterization of −α2

n asserts that∫ h

0

(|φ′2l −K(y)|φl|2)dy > −α2n

∫ h

0

|φl|2dy.

Accordingly, ∫ h0

0

(l2α2 − α2n)|φl|2dy 6 0

must hold. Since α > αmax/2 and l > 2, it follows that l2α2−αn > 2α2max−α2

n > 0.Consequently, φl ≡ 0.

Therefore, ψ∞ ≡ 0, which contradicts since ‖φ∞‖L2 = 1. This completes theproof. ¤

Lemma 5.6. Under the assumption of Lemma 5.5, for any b ∈ L2(Sε), there existsa unique solution ψb to (5.7). Moreover, the estimate

(5.17) ‖ψb‖H1(Dε) 6 C‖b‖L2(Sε)

holds, where C > 0 is independent of ε, b.

Proof. The proof uses the theory of Fredholm alternative as adapted to usual el-liptical problems [22, Section 6.2]. Let us introduce the Hilbert space

(5.18) H (Dε) = ψ ∈ H1(Dε) : ψ(x, 0) = 0and a bilinear form Bz : H ×H → R, defined as

Bz[φ, ψ] =∫∫

Dε

(∇φ · ∇ψ∗ + φ(Kλψ)∗)dydx + z (φ, ψ) .

Here,

Kλψ = −γ(ψε)ψ + γ′(ψε)∫ 0

−∞λeλsψ(Xε(s), Yε(s))ds,

z ∈ R and (·, ·) denotes the L2 (Dε) inner product. By the estimate (5.15) it followsthat

(5.19) |Bz[φ, ψ]| 6 (C(γ, δ) + z)‖φ‖H1‖ψ‖H1

and

(5.20) Bz[φ, φ] > c0‖φ‖2H1

for z > 2C(γ, δ), where

C(γ, δ) = ‖γ′(ψε)‖L∞

(1 +

|λ0|+ δ

δ

)

and c0 = min(1, C(γ, δ)) > 0. Then, the Lax-Milgram theorem applies and thereexists a bounded operator Lz : H∗ → H such that Bz[Lzf, φ] = 〈f, φ〉 for anyf ∈ H∗ and φ ∈ H, where H∗ denotes the dual space of H and 〈·, ·〉 is the dualitypairing. For b ∈ L2(Sε), the trace theorem [22, Section 5.5] permits us to defineb∗ ∈ H∗ by

< b∗, ψ >=∮

Sε

b(x)ψ∗(x, ηε(x))dx for ψ ∈ H.


Note that ψb ∈ H is a weak solution of (5.7) if and only if

Bz(ψb, φ) =< zψb + b∗, φ > for all φ ∈ H.

That is to say, ψb = Lz(zψb + b∗), or equivalently

(5.21) (id− zLz)ψb = Lzb∗.

The operator Lz : L2(Dε) → L2(Dε) is compact. Indeed,

‖Lzφ‖H1(Dε)6 ‖Lz‖H∗→H ‖φ‖H∗ 6 C ‖φ‖L2(Dε)

for any φ ∈ L2(Dε). Moreover, the result of Lemma 5.5 states that ker(I − zLz) =0. Thus, by the Fredholm alternative theory for compact operators, the equation(5.21) is uniquely solvable for any b ∈ L2(Sε) and

(5.22) ψb = (id− zLz)−1

Lzb∗.

Next is the proof of (5.17). From (5.22) it follows that

‖ψb‖L2(Dε) 6∥∥(id− zLz)−1

∥∥L2→L2 ‖Lz‖H∗→H ‖b∗‖H∗ 6 C‖b‖L2(Sε),

where C > 0 is independent of b and ε. Then, by (5.21) it follows that

‖ψb‖H1(Dε)= ‖Lz(zψb) + Lzb

∗‖H 6 C(‖ψb‖L2(Dε) + ‖b‖L2(Sε))

6 C ′‖b‖L2(Sε),

where C, C ′ > 0 are independent of b and ε. This completes the proof. ¤

Similar considerations to the above proves the unique solvability of the inhomo-geneous problem.

Corollary 5.7. Under the assumption of Lemma 5.5, for any f ∈ H∗(Dε) thereexists an unique weak solution ψf ∈ H1(Dε) to the elliptic problem

∆ψ + γ′(ψε)ψ − γ′(ψε)∫ 0

−∞λeλsψ(Xε(s), Yε(s))ds = f in Dε,

ψn = 0 on Sε,

ψ(x, 0) = 0.

The estimate

(5.23) ‖ψf‖H1(Dε) 6 C‖f‖H∗(Dε)

holds, where C > 0 is independent of f and ε. Moreover, if f ∈ L2(Dε) then animproved estimate

(5.24) ‖ψf‖H2(Dε) 6 C‖f‖L2(Dε)

holds, where C > 0 is independent of f and ε.

Proof. As in the proof of Lemma 5.6, the function ψf is a solution to the aboveboundary value problem if and only if

(id− zLz)ψf = Lzf.

The unique solvability and the H1-estimate (5.23) are identically the same as thosein Lemma 5.6. When f ∈ L2(Dε) the elliptic regularity theory [2] for Neumannboundary condition implies (5.24). ¤


For any b ∈ L2per(Sε) let us define the operator

(5.25) Tεb = ψb|Sε = ψb (x, ηε(x)) ,

where ψb is the unique solution of (5.7) in Lemma 5.6 with periodicity. Then, theelliptic estimate (5.17) and the trace theorem [22, Section 5.5] implies that

(5.26) ‖Tεb‖H1/2(Sε) 6 C‖ψb‖H1(Dε) 6 C ′‖b‖L2 ;

Therefore, Tε : L2per(Sε) → L2

per(Sε) is a compact operator.

5.3. Proof of Theorem 5.1. This subsection is devoted to the proof of Theorem5.1 pertaining to the linear instability of small-amplitude periodic traveling wavesover an unstable shear flow.

For ε > 0 sufficiently small and |λ− λ0| 6 (Re λ0)/2, let us denote

(5.27) F(λ, ε) = T (λ, ε) C (λ, ε) (PεyC (λ, ε) + Ω id) : L2per(Sε) → L2

per(Sε),

where C(λ, ε) = Cλ and T (λ, ε) = Tε are defined in (5.5) and (5.25), respectively.In the light of the discussion in the previous subsection, it suffices to show that foreach small parameter ε > 0 there exists λ(ε) with |λ(ε)−λ0| 6 (Re λ0)/2 such thatthe operator id +F(λ(ε), ε) has a nontrivial null space. The result of Theorem 4.2states that there exists λ0 = −iαcα with Im cα > 0 such that id + F(λ0, 0) has anontrivial null space. The proof for ε > 0 uses a perturbation argument, based onthe following lemma due to Steinberg [41].

Lemma 5.8. Let F (λ, ε) be a family of compact operators on a Banach space,analytic in λ in a region Λ in the complex plane and jointly continuous in (λ, ε) foreach (λ, ε) ∈ Λ × R. Suppose that id − F (λ0, ε) is invertible for some λ0 ∈ Λ andall ε ∈ R. Then, R(λ, ε) = (id − F (λ, ε))−1 is meromorphic in Λ for each ε ∈ Rand jointly continuous at (z0, ε0) if λ0 is not a pole of R(λ, ε); its poles dependcontinuously on ε and can appear or disappear only at the boundary of Λ.

In order to apply Lemma 5.8 to our situation, we need to transform the operator(5.27) to one on a function space independent of the parameter ε. This calls for theemployment of the action-angle mapping Aε, as is done in the proof of Lemma 5.5:

Aε : Dε → D and Aε(x, y) = (θ, I),

where the action-angle variables (θ, I) are defined in (5.13) and

D = (θ, I) : 0 < θ < 2π/α , 0 < I < h0.Note that Aε maps Sε bijectively to (θ, h0) : 0 < θ < 2π/α. The latter may beidentified with (2π/α, h0). This naturally induces an homeomorphism

Bε : L2per(Sε) → L2

per([0, 2π/α])

by(Bεf)(θ) = f(A−1

ε (θ, h0)).Let us denote the following operators from L2

per ([0, 2π/α]) to itself:

T (λ, ε) = BεT ((λ, ε)) (Bε)−1,

C (λ, ε) = BεC (λ, ε) (Bε)−1,

Pε = BεPεy(Bε)−1,

and

(5.28) F(λ, ε) = BεF(λ, ε)(Bε)−1 = T (λ, ε) C(λ, ε)(PεC (λ, ε) + Ωid).


Since T (λ, ε) is compact and T (λ, ε) and C(λ, ε) are analytic in λ with |λ−λ0| 6(Re λ0)/2, the operator F(λ, ε) is compact and analytic in λ. Subsequently, F(λ, ε)is compact and analytic in λ. Clearly, Pεy(x), C(λ, ε) and Bε are continuous inε, and in turn, C(λ, ε) and Pε are continuous in ε. Thus, it remains to show thecontinuity of T (λ, ε) in ε to obtain the the continuity of F(λ, ε) in ε.

Lemma 5.9. For ε > 0 sufficiently small and |λ − λ0| 6 (Re λ0)/2, the operatorT (λ, ε) satisfies the estimate

(5.29) ‖T (λ, ε1)− T (λ, ε2)‖L2per([0,2π/α])→L2

per([0,2π/α]) 6 C|ε1 − ε2|,

where C > 0 is independent of λ and ε.

Proof. For a given b ∈ L2per([0, 2π/α]), let us denote bε = B−1

ε b ∈ L2(Sε). Bydefinition

T (λ, ε) b = BεTεbε = Bε(ψb,ε|Sε),

where ψb,ε ∈ H1 (Dε) is the unique weak solution of

∆ψb,ε + γ′(ψε)ψb,ε − γ′(ψε)∫ 0

−∞λeλsψb,ε(Xε(s), Yε(s))ds = 0 in Dε;(5.30a)

(ψb,ε)n = bε on Sε;(5.30b)

ψb,ε(x, 0) = 0.(5.30c)

Our goal is to estimate the L2-operator norm of T (λ, ε1) − T (λ, ε2) in termsof |ε1 − ε2|. Since the domain of the boundary value problem (5.30) depends onε, we use the action-angle mapping Aεj (j = 1, 2) to transform functions andthe Laplacian operator in Dεj (j = 1, 2) to those in the fixed domain D. For thesimplicity of notation, we writeAεj (j = 1, 2) to denote the induced transformationsfor functions and operators. Let ψj = Aεj (ψb,εj ), which are H1(D)-functions, andlet

∆j = Aεj (∆), γj(I) = Aεj (γ′(ψεj )),

where j = 1, 2. By definition, Aεj (bεj ) = Bεj (bεj ) = b. Note that the characteristicequation (3.7) in the action-angle variables (θ, I) becomes

θ = υj(I)I = 0

for j = 1, 2, where υj(I) = υεj (I). That means, the trajectory (Xεj (s), Yεj (s)) inthe phase space transforms under the mapping Aεj into (θ + υj(I)s, I). Since ψj

(j = 1, 2) are 2π/α-periodic in θ, it is equipped with the Fourier expansion

ψj(θ, I) =∑

l

eilθψj,l(I).


Under the action-angle mapping Aεj (j = 1, 2) the left side of (5.30) becomes

Aεj

(γ′(ψε)ψb,ε − γ′(ψε)

∫ 0

−∞λeλsψb,ε(Xε(s), Yε(s))ds

)

= γj(I)

(∑

l

eilθψj,l(I)−∫ 0

−∞λeλs

∑

l

eil(θ+υj(I)s)ψj,l(I)ds

)(5.31)

= γj(I)∑

l

ilυj(I)λ + ilυj(I)

ψj,l(I)eilθ,

and thus the system (5.30) becomes

∆jψj + γj(I)∑

l

ilυj(I)λ + ilυj(I)

ψj,l(I)eilθ = 0 in D,

∂Iψj(θ, h0) = b(θ),

ψj(θ, 0) = 0.

Accordingly, the difference ψ1 − ψ2 is a weak solution of the partial differentialequation

∆1(ψ1 − ψ2) + (∆1 −∆2)ψ2 + γ1

∑

l

ilυ1

λ + ilυ1(ψ1,l(I)− ψ2,l(I))eilθ

+ γ1

∑

l

(ilυ1

λ + ilυ1− ilυ2

λ + ilυ2

)ψ2,l(I)eilθ + (γ1 − γ2)

∑

l

ilυ2

λ + ilυ2ψ2,l(I)eilθ = 0

(5.32)

with the boundary conditions

∂I(ψ1 − ψ2)(θ, h0) = 0,(5.33a)

(ψ1 − ψ2)(θ, 0) = 0.(5.33b)

Let us write (5.32) as

(5.34) ∆1(ψ1 − ψ2) + γ1

∑

l

ilυ1(I)λ− ikυ1(I)

(ψ1,l(I)− ψ2,l(I))eilθ = f,

where f = f1 + f2 + f3 and

f1 = −(∆1 −∆2)ψ2,

f2 = −γ1

∑

l

iλl(υ1 − υ2)(λ + ilυ1)(λ + ilυ2)

ψ2,l(I)eilθ,

f3 = −(γ1 − γ2)∑

l

ilυ2

λ + ilυ2ψ2,l(I)eilθ.

We estimate f1, f2, f3 separately. To simplify notations, C > 0 in the estimatesbelow denotes a generic constant independent of ε and λ.

First, we claim that f1 ∈ H∗(D) with the estimates

(5.35) ‖f1‖H∗(D0)6 C|ε1 − ε2|‖b‖L2([0,2π/α]),

where H∗(D) is the dual space of

H(D) = ψ ∈ H1(D) : ψ(θ, I) = ψ(θ + 2π/α, I), ψ(θ, 0) = 0.


Let us write

∆j = ajII∂II + aj

Iθ∂Iθ + ajθθ∂θθ + bj

I∂I + bjθ∂θ for j = 1, 2,

and the difference of the coefficients as

aII = a1II − a2

II , aIθ = a1Iθ − a2

Iθ, aθθ = a1θθ − a2

θθ,

bI = b1I − b2

I , and bθ = b1θ − b2

θ.

Then formally, for any φ ∈ H(D) ∩ C2(D) it follows that∫

D0

f1φ dIdθ(5.36)

=∫

D0

−φ(aII∂II + aIθ∂Iθ + aθθ∂θθ + bθ∂θ + bI∂I

)ψ2dIdθ

=∫

D0

[∂Iψ2∂I (aIIφ) + ∂Iψ2∂θ (aIθφ) + ∂θψ2∂θ (aθθφ)] dIdθ

−∫

D0

φ(bI∂Iψ2 + bθ∂θψ2

)dIdθ −

∫

I=h0aIIφb (θ) dθ,

This uses that ψ2 and φ are periodic in the θ-variable and that

∂Iψ2(θ, h0) = b(θ), φ(θ, 0) = 0.

Note that the elliptic estimate (5.17) and the equivalence of norms under the trans-formation A−1

ε2 assert that

‖ψ2‖H1(D) 6 C∥∥A−1

ε2 ψ2

∥∥H1(Dε2 )

= C ‖ψb,ε2‖H1(Dε2 )

6 C ‖bε2‖L2(Sε2 ) 6 C ‖b‖L2([0,2π/α]) .

Since|aII |C1 + |aIθ|C1 + |aθθ|C1 +

∣∣bθ

∣∣C1 +

∣∣bI

∣∣C1 = O(|ε1 − ε2|),

by using the trace theorem it follow from (5.36) the estimate∣∣∣∣∫

D

f1φ dIdθ

∣∣∣∣ 6 C |ε1 − ε2|(‖ψ2‖H1(D) + ‖b‖L2([0,2π/α])

)‖φ‖H1(D)

6 C |ε1 − ε2| ‖b‖L2([0,2π/α]) ‖φ‖H1(D) .

This proves the estimate (5.35).We claim that if b is smooth then the formal manipulations in (5.36) are valid and

ψ2 ∈ C2(D). Note that Theorem 2.2 ensures that the steady state (ηε2(x), ψε2(x, y))is in C3+β class, where β ∈ (0, 1). Since b is smooth it follows that bε2 = B−1

ε2 b is atleast in H2 (Sε2). Then, the similar argument as in the regularity proof of Theorem5.1 asserts that ψb,ε2 ∈ H7/2(Dε2) ⊂ C2(Dε2). Since the definition of the action-angle variables guarantees that the mapping Aε2 is at least of C2, subsequently,ψ2 = Aε2ψb,ε2 ∈ C2(D). This proves the claim. If b ∈ L2, an approximation of bby smooth functions asserts (5.35). This proves the claim.

Next, since∣∣∣∣1

λ + ilυj

∣∣∣∣ =1

(|Re λ|2 + |Im λ− lυj |2)1/26 1|Re λ| 6 2

Re λ0,

by the estimate (5.17) it follows that

(5.37) ‖f2‖L2(D) 6 C|ε1 − ε2|‖ψ2‖L2(D) 6 C|ε1 − ε2|‖b‖L2([0,2π/α]).


Similarly,

(5.38) ‖f3‖L2(D0)6 C|ε1 − ε2|‖b‖L2([0,2π/α]).

Combining the estimates (5.35), (5.37) and (5.38) asserts that f ∈ H∗(D) and

‖f‖H∗(D) 6 C|ε1 − ε2|‖b‖L2([0,2π/α]).

Let ψ = ψ1 − ψ2 ∈ H1(D) and φ = A−1ε1 ψ ∈ H1 (Dε1). It remains to transform

back to the physical space of the boundary value problem for ψ and to compute theoperator norm of T (λ, ε1)−T (λ, ε2). Under the transformation A−1

ε1 , the equations(5.34), (5.33a), (5.33b) become

∆φ + γ′(ψε1)φ− γ′(ψε1)∫ 0

−∞λeλsφ(Xε1(s), Yε1(s))ds = A−1

ε1 f in Dε1 ;

(φε1)n = 0 on Sε1 ;

φε1(x, 0) = 0,

Then, A−1ε1 f ∈ H∗(Dε1) and

∥∥A−1ε1 f

∥∥H∗(Dε1 )

6 C ‖f‖H∗(D) 6 C|ε1 − ε2|‖b‖L2([0,2π/α]).

Corollary 5.7 thus applies to assert that

‖ψ‖H1(D) 6 ‖φ‖H1(Dε1 ) 6 C∥∥A−1

ε1 f∥∥

H∗(Dε1 )

6 C|ε1 − ε2|‖b‖L2([0,2π/α]).

Finally, by the trace theorem it follows

‖T (λ, ε1) b− T (λ, ε2) b‖L2per([0,2π/α]) = ‖(ψ1 − ψ2) (θ, h0)‖L2

per([0,2π/α])

6 C ‖ψ‖H1(D) 6 C|ε1 − ε2|‖b‖L2per([0,2π/α]).

This completes the proof. ¤

We are now in a position to prove our main theorem.

Proof of Theorem 5.1. For |λ − λ0| 6 (Re λ0)/2, where λ0 = −iαcα, and ε > 0small, consider the family of operators F(λ, ε) on L2

per([0, 2π/α]), defined in (5.28).The discussions following (5.28) and Lemma 5.9 assert that F(λ, ε) is compact,analytic in λ and continuous in ε.

By assumption, Im cα > 0 and cα is an unstable eigenvalue of the Rayleighsystem (4.1)-(4.2) which corresponds to ε = 0. In other words, λ0 is a pole of(id + F(λ, 0))−1. Subsequently, it is a pole of (id + F(λ, 0))−1. Since λ0 is anisolated pole, we may choose δ > 0 small enough so that the operator id + F(λ, 0)is invertible on |λ−λ0| = δ. By the continuity of F(λ, ε) in ε, the following estimate

‖F(λ, ε)− F(λ, 0)‖L2per([0,2π/α])→L2

per([0,2π/α]) 6 Cε

holds. Hence, id + F(λ, ε) is invertible on |λ− λ0| = δ and ε > 0 sufficiently small.Lemma 5.8 then applies the poles of (id + F(λ, ε))−1 are continuous in ε and canonly appear or disappear in the boundary of λ : |λ− λ0 < δ. Therefore, at eachε > 0, there is found a pole λ(ε) of id+F(λ, ε) in |λ(ε)−λ0| < δ. Then, Re λ(ε) > 0and there exists a nonzero function f ∈ L2

per([0, 2π/α]) such that

(id + F(λ, ε))f = 0.


Our next task is to construct an exponentially growing solution to the linearizedsystem (3.5) associated to f . Let f = (Bε)−1f be the function in L2

per(Sε) whichtransforms to f under the action-angle mapping. By the construction of F , itfollows that

(5.39) (id + F(λ, ε))f = 0

Given f ∈ L2(Sε) let ψ ∈ H1(Dε) be the unique solution of (5.7) with λ = λ(ε) and

ψn(x) = −Cλ(Pεy(x)Cλ + Ω id)f(x),

and let η ∈ L2(Sε) be defined as η(x) = Cλf(x). Let

ψ(x, ηε(x)) = −Tε(Cλ(Pεy(x)Cλ + Ω id))f(x).

This yields

ψn(x) = −Cλ (P (x, ηε(x)) + Ωψ(x, ηε(x))) ,(5.40)

η(x) = Cλ(ψ(x, ηε(x)).(5.41)

We shall show that (eλ(ε)tη(x), eλ(ε)tψ(x, y)) satisfies the linearized system (3.5).Since ψ(x, 0) = 0, the bottom boundary condition (3.5e) is satisfied. It is readilyseen the dynamic boundary condition (3.5c) is satisfied. In view of Lemma 5.3,the equations (5.41), (5.40) imply that η and ψn(x) satisfy (5.1b) and (5.1d), re-spectively, in the weak sense. In turn, the boundary conditions on the top surface(3.5b) and (3.5d) are satisfied in the weak sense. Since ψ solves (5.7a), the vorticityis given as

(5.42) ω = −∆ψ = γ′(ψε)ψ − γ′(ψε)∫ 0

−∞λeλsψ(Xε(s), Yε(s))ds.

As shown in [35], above equation implies that the vorticity ω satisfies the thevorticity equation (3.6) in the weak sense. In other words, ψ satisfies (3.5a) inthe weak sense. In summary, (eλ(ε)tη(x), eλ(ε)tψ(x, y)) is a weak solution of thelinearized system (3.5).

Our last step of the proof is to study the regularity of the growing-mode (eλ(ε)tη(x),eλ(ε)tψ(x, y)) and thus to show the solvability in the strong sense. First, we claimthat ψ ∈ H2 (Dε). (Recall that the result of Lemma 5.6 is ψ ∈ H1 (Dε).) Indeed,the trace theorem asserts that ψ ∈ H1 (Dε) implies ψ(x, ηε(x)) ∈ H1/2 (Sε). Sincethe operator Cλ defined by (5.5) is regularity preserving, by (5.40) it follows thatψn(x) ∈ H1/2 (Sε). Recall from the statement of Theorem 2.2 that ηε is of C3+β(R),where β ∈ (0, 1). That is Dε is of C3+β . Since ω = −∆ψ ∈ L2 (Dε) from the regu-larity theorem ([2]) of elliptic boundary problems it follows that ψ ∈ H2(Dε). Then,by the trace theorem and (5.40), respectively, follows that ψ(x, ηε(x)) ∈ H3/2(Sε)and ψn(x) ∈ H3/2(Sε).

In order to obtain a higher regularity of ψ, we need to show that ω ∈ H1(Dε).The argument presented below is a simpler version of that in [33]. Taking thegradient of (5.42) yields that

∇ω =∇(γ′(ψε))ψ + γ′(ψε)∇ψ −∇(γ′(ψε))∫ 0

−∞λeλsψ(Xε(s), Yε(s))ds

− γ′(ψε)∫ 0

−∞λeλs∇ψ(Xε(s), Yε(s))

∂(Xε(s), Yε(s))∂(x, y)

ds.

(5.43)


Note that the particle trajectory is written in the action-angle variables (θ, I) =Aε(x, y) as

(Xε(s;x, y), Yε(s; x, y)) = A−1ε ((θ + υε(I)s, I))

This relies on that the action-angle mapping Aε is globally defined, a consequenceof the assumption of no stagnation. With the use of the above description of thetrajectory the estimate of the Jacobi matrix

(5.44)∣∣∣∣∂ (Xε(s; x, y), Yε(s; x, y))

∂ (x, y)

∣∣∣∣ 6 C1 |s|+ C2

follows, where C1, C2 > 0 are independent of s.It is straightforward to see that calculations as in the proof of (5.15), L2-norm

of the first three terms of (5.43) is bounded by the H1 norm of ψ. The last termin (5.43) is treated as

∥∥∥γ′(ψε)∫ 0

−∞λeλs∇ψ(Xε(s), Yε(s))

∂(Xε(s), Yε(s))∂(x, y)

ds∥∥∥

L2

6 ||γ′(ψε)||L∞∫ 0

−∞|λ|eRe λs(C1 |s|+ C2)||∇ψ(Xε(s), Yε(s))||L2(Dε)ds

6 C ‖ψ‖H1(Dε).

This uses (5.44), Re λ > δ > 0 and that the mapping (x, y) 7→ (Xε(s), Yε(s)) ismeasure-preserving. Therefore,

‖∇ω‖L2(Dε) 6 C ‖ψ‖H1(Dε).

In turn, ω ∈ H1(Dε). Since ψn(x) ∈ H3/2(Sε), by the elliptical regularity theorem[2] it follows that that ψ ∈ H3 (Dε). In view of the trace theorem this impliesψn ∈ H5/2(Sε).

We repeat the process again. Taking the gradient of (5.43) and using the linearstretching property (5.3) of the trajectory, it follows that ω ∈ H2 (Dε). The ellipticregularity applies to assert that ψ ∈ H4(Dε) ⊂ C2+β(Dε), where β ∈ (0, 1). By thetrace theorem then it follows that ψ(x, ηε(x)) ∈ H7/2(Sε). On account of (5.41) thisimplies that η ∈ H7/2 (Sε) ⊂ C2+β([0, 2π/α]). Therefore, (eλ(ε)tη(x), eλ(ε)tψ(x, y))is a classical solution of (3.5). This completes the proof. ¤

6. Instability of general shear flows

Linear instability of free-surface shear flows is of independent interests. Thissection extends our instability result in Theorem 4.2 to a more general class ofshear flows. The following class of flows was introduced in [32] and [35] in therigid-wall setting.

Definition 6.1. A function U ∈ C2([0, h]) is said to be in the class F if U ′′ takesthe same sign at all points such that U(y) = c, where c in the range of U but notan inflection value of U .

Examples of the class-F flows include monotone flows and symmetric flows witha monotone half. If U ′′(y) = f(U(y))k(y) for f continuous and k(y) > 0 then U isin class F . All flows in class K+ are in class F .

The lemma below shows that for a flow in class F a neutral limiting wave speedmust be an inflection value. The main difference of the proof from that in class K+


(Proposition 4.4) lies in the lack of uniform H2-bounds of unstable solutions neara neutral limiting mode.

Lemma 6.2. For U ∈ F , let (φk, αk, ck)∞k=1 with Im ck > 0 be a sequence ofunstable solutions which satisfy (4.1)–(4.2). If (αk, ck) converges to (αs, cs) ask →∞ with αs > 0 and cs in the range of U then cs must be an inflection value ofU .

Proof. Suppose on the contrary that cs is not an inflection value. Let y1, y2, . . . , ym

be in the pre-image of cs so that U(yj) = cs, and let S0 be the complement ofthe set of points y1, y2, . . . , ym in the interval [0, h]. Since cs is not an inflectionvalue, Definition 6.1 asserts that U ′′(yj) takes the same sign for j = 1, 2, . . . , m,say positive. As in the proof of Proposition 4.4, let Eδ = y ∈ [0, h] : |y − yj | <δ for some j, where j = 1, 2, · · · ,m. It is readily seen that Ec

δ ⊂ S0. Note thatU ′′(y) > 0 for y ∈ Eδ if δ > 0 small enough. After normalization, ‖φk‖L2 = 1. Theresult of Lemma 3.6 in [32] is that φk converges uniformly to φs on any compactsubset of S0. Moreover, φ′′s exists on S0 and φs satisfies

φ′′s − α2sφs − U ′′

U − csφs = 0 for y ∈ (0, h).

Our first task is to show that φs is not identically zero. Suppose otherwise. Theproof is again divided into two cases.

Case 1: U(h) 6= cs. In this case, [h− δ1, h] ⊂ S0 for some δ1 > 0. As is done inthe Proposition 4.4, for any q real follows that

∫ h

0

(|φ′k|2 + α2

k|φk|2 +U ′′(U − q)|U − ck|2 |φk|2

)dy

>∫ h

0

|φ′k|2 dy + α2k +

∫

Ecδ

U ′′(U − q)|U − ck|2 |φk|2dy +

∫

Eδ

U ′′(U − q)|U − ck|2 |φk|2dy

>∫ h

0

|φ′k|2 dy + α2k − sup

Ecδ

|U ′′(U − q)||U − ck|2

∫

Ecδ

|φk|2dy.

On the other hand, (4.10) with q = Umin − 1 yields that∫ h

0

(|φ′k|2 + α2

k|φk|2 +U ′′(U − Umin + 1)

|U − ck|2 |φk|2)

dy

=(

Re gr(ck) + (Re ck − Umin + 1)Im gr(ck)

Im ck

)|φk(h)|2

6 C|φk(h)|2 6 C1

(ε

∫ h

h−δ1

|φ′2k dy +1ε

∫ h

h−δ1

|φk|2dy

).

If ε is chosen to be small then the above two inequalities lead to

0 > α2k − sup

Ecδ

|U ′′(U − Umin + 1)||U − ck|2

∫

Ecδ

|φk|2dy − Cε

∫ h

h−δ1

|φk|2dy.

Since φk converges to φs ≡ 0 uniformly on Ecδ and [h−δ1, h], this implies 0 > α2

s/2.A contradiction proves that φs is not identically zero.

Case 2: U(h) = cs. Since

Im ck

∫ h

0

U ′′

|U − ck|2 |φk|2dy = −Im gs(ck) |φ′k (h)|2 ,


the identity (4.17) yields that∫ h

0

(|φ′′k |2 + 2α2k |φ′k|2 + α4

k|φk|2)dy

= −2α2k Re gs(ck) |φ′k (h)|2 +

∫ h

0

(U ′′)2

|U − ck|2 |φk|2dy

=(−2α2

k Re gs(ck)− Im gs(ck)Im ck

)|φ′k (h)|2

6 Cd(ck, U(h)) |φ′k (h)|2 6 Cd(ck, U(h))‖φk‖2H2 .

Here, d(ck, U(h)) = |Re ck − U(h)| + (Im ck)2. Since d(ck, U(h)) → 0 as k → ∞,this implies that 0 > α4

s/4. A contradiction proves that φs is not identically zero.Subsequently, Lemma 4.7 asserts that φs(yj) 6= 0 for some yj .

It remains to prove that cs is the inflection value of U . In Case 1 when U(h) 6= cs,it is straightforward to see that

∫

Eδ

U ′′(U − Umin + 1)|U − cs|2 |φs|2dy >

∫

|y−yj |<δ

U ′′

|U − cs|2 |φs|2dy = ∞,

where φs(yj) 6= 0. As in the proof of Proposition 4.4, Fatou’s lemma then statesthat

lim infk→∞

∫

Eδ

U ′′(U − Umin + 1)|U − ck|2 |φk|2dy = ∞.

Subsequently, (4.10) applies and

0 =∫ h

0

(|φ′k|2 + α2

k|φk|2 +U ′′(U − Umin + 1)

|U − c|2 |φk|2)

dy

−(

Re gr(ck) + (Re ck − Umin + 1)Im gr(ck)

Im ck

)|φk(h)|2

>∫

Eδ

U ′′(U − Umin + 1)|U − ck|2 |φk|2dy − sup

Ecδ

|U ′′(U − Umin + 1)||U − ck|2 − Cε > 0

for k large. A contradiction asserts that cs is an inflection value. A similar consid-eration in Case 2 when U(h) = cs states that

lim infk→∞

∫

Eδ

U ′′(U ′′max + 1− U ′′)|U − ck|2 |φk|2dy = ∞,

where U ′′max = max[0,h] U

′′(y). On the other hand, (4.17) implies

0 >∫

Eδ

U ′′(U ′′max + 1− U ′′)|U − ck|2 |φk|2dy − sup

Ecδ

|U ′′(U ′′max + 1− U ′′)||U − ck|2 > 0

for k large. A contradiction asserts that cs is an inflection value. This completesthe proof. ¤

The proof of above lemma indicates that a flow in class F is linearly stable whenthe wave number is large.

Lemma 6.3. For U ∈ F there exists αmax > 0 such that for each α > αmax

corresponds no unstable solution to (4.1)–(4.2).


Proof. Suppose otherwise. Then, there would exist a sequence of unstable solutions(φk, αk, ck)∞k=1 of (4.1)–(4.2) such that αk →∞ as k →∞. After normalization,let ‖φk‖L2 = 1. Our goal is to show that limk→∞ Im ck = 0. If Im ck > δ > 0for some δ then 1/ |U − ck| and |gr(ck)| are uniformly bounded. Accordingly, withq = Re ck in (4.10) yields

0 =∫ h

0

(|φ′k|2 + α2

k|φk|2 +U ′′(U − Re ck)|U − ck|2 |φk|2

)dy − Re gr(ck)|φk(h)|2

>∫ h

0

|φ′k|2 dy + α2k − sup

[0,h]

|U ′′(U − Re ck)||U − ck|2 − C

(ε

∫ h

0

|φ′k|2 dy +1ε

∫ ε

0

|φk|2dy

)

> α2k − sup

[0,h]

|U ′′(U − Re ck)||U − ck|2 − C/ε > 0

for k sufficiently large. A contradiction proves that ck → cs ∈ [Umin, Umax] ask →∞. The remainder of the

proof is nearly identical to that of Lemma 6.2 and hence is omitted. ¤

The following theorem gives a necessary condition for free surface instability thatthe flow profile should have an inflection point, which generalize the classical resultof Lord Rayleigh [42] in the rigid wall case.

Theorem 6.4. A flow U ∈ F without an inflection point throughout [0, h] is linearlystable.

Proof. Suppose otherwise; Then, there would exist an unstable solution (φ, α, c) to(4.1)–(4.2) with α > 0 and Im c > 0. Lemma 4.8 allows us to continue this unstablemode for wave numbers to the right of α until the growth rate becomes zero. Notethat a flow without an inflection point is trivially in class F . So by Lemma 6.3,this continuation must end at a finite wave number αmax and a neutral limitingmode therein. On the other hand, Lemma 6.2 asserts that the neutral limitingwave speed cs corresponding to this neutral limiting mode must be an inflectionvalue. A contradiction proves the assertion. ¤

Remark 6.5. Our proof of the above no-inflection stability theorem is very differentfrom the rigid wall case. In the rigid-wall setting, where φ(h) = 0, the identity(4.14) reduces to

ci

∫ h

0

U ′′

|U − c|2 |φ|2dy = 0,

which immediately shows that if U is unstable (ci > 0) then U ′′(y) = 0 at somepoint y ∈ (0, h). The same argument was adapted in [48, Section 5] for the free-surface setting, however, it does not give linear stability for general flows with noinflection points. More specifically, in the free-surface setting, (4.14) becomes

ci

∫ h

0

U ′′

|U − c|2 |φ|2dy =

(2g(U(h)− cr)|U(h)− c|4 +

U ′(h)|U(h)− c|2

)|φ(h)|2 ,

which only implies linear stability ([48, Section 5]) for special flows satisfyingU ′′ (y) < 0, U ′ (y) > 0 or U ′′ (y) > 0, U ′ (y) 6 0. In the proof of Theorem 6.4,we use the characterization of neutral limiting modes and remove above additionalassumptions.


Let us now consider a shear flow U ∈ F with multiple inflection values U1, U2, . . . , Un.Lemma 6.2 states that a neutral limiting wave speed cs must be one of the inflectionvalues U1, U2, . . . , Un, say cs = Uj . Repeating the argument in the proof of Lemma4.6 around inflection points corresponding to the inflection value Uj establishes anuniform H2-bound for a sequence of unstable solutions near the neutral limitingmode whose wave speed is near Uj . The proof is very similar to that in the rigid-wall setting [35, Lemma 2.6] and hence it is omitted. Thus, neutral limiting modesin class F are characterized by inflection values.

Proposition 6.6. If U ∈ F has inflection values U1, U2, . . . , Un then for a neutrallimiting mode (φs, αs, cs) with αs > 0 the neutral limiting wave speed must be oneof the inflection values, that is, cs = Uj for some j. Moreover, φs must solve

φ′′s − α2sφs + Kj(y)φs = 0 for y ∈ (0, h),

where Kj(y) = − U ′′(y)U(y)− Uj

, and boundary conditions

(6.1)

φ′s(h) = gr(Uj), φs(0) = 0 if U(h) 6= Uj

φs(h) = 0, φs(0) = 0 if U(h) = Uj .

One may exploit the instability analysis of Theorem 4.9 for a flow in class F withpossibly multiple inflection values). The main difference of the analysis in class Ffrom that in class K+ is that unstable wave numbers in class F may bifurcate tothe left and to the right of a neutral limiting wave number, whereas unstable wavenumbers in class K+ bifurcate only to the left of a neutral limiting wave number. Inthe rigid-wall setting, with an extension of the proof of [32, Theorem 1.1] Zhiwu Lin[35, Theorem 2.7] analyzed this more complicated structure of the set of unstablewave numbers. The remainder of this section establishes an analogous result in thefree-surface setting.

In order to study the structure of unstable wave numbers in class F with possiblymultiple inflection values, we need several notations to describe. A flow U ∈ F issaid to be in class F+ if each Kj(y) = −U ′′(y)/(U(y) − Uj) is nonzero, where Uj

for j = 1, · · · , n are inflection values of U . It is readily seen that for such a flowKj takes the same sign at all inflection points of Uj . A neutral limiting mode(φj , αj , Uj) is said to be positive if the sign of Kj is positive at inflection points ofUj , and negative if the sign of Kj is negative. Proposition 6.6 asserts that −α2

s is anegative eigenvalue of − d2

dy2 −Kj(y) on y ∈ (0, h) with boundary conditions (6.1).We employ the argument in the proof of Theorem 4.9 to conclude that an unstablesolution exists near a positive (negative) neutral limiting mode if and only if theperturbed wave number is slightly to the left (right) of the neutral limiting wavenumber. Thus, the structure of the set of unstable wave numbers with multipleinflection values is more intricate. We remark that a class-K+ flow has a uniquepositive neutral limiting mode and hence unstable solutions bifurcate to the left ofa neutral limiting wave number.

Let us list all neutral limiting wave numbers in the increasing order. If thesequence contains more than one successive negative neutral limiting wave numbers,then we pick the smallest (and discard others). If the sequence contains more thanone successive positive neutral limiting wave numbers, then we pick the largest(and discard others). If the smallest member in this sequence is a positive neutrallimiting wave number, then we add zero into the sequence. Thus, we obtain a new


sequence of neutral limiting wave numbers. Let us denote the resulting sequence byα−0 < a+

0 < · · · < α−N < α+N , where, α−0 (might be 0),. . . , α−N are negative neutral

limiting wave numbers and α+0 , . . . , α+

N are positive neutral limiting wave numbers.The largest member of the sequence must be a positive neutral wave number sinceno unstable modes exist to its right.

Theorem 6.7. For U ∈ F+ with inflection values U1, U2, . . . , Un, let α−0 < a+0 <

· · · < α−N < α+N be defined as above. For each α ∈ ∪N

j=0(α−j , α+

j ), there existsan unstable solution of (4.1)–(4.2. Moreover, the flow is linear stable if eitherα > α+

N or all operators − d2

dy2 −Kj(y) (j = 1, 2, . . . , n) on y ∈ (0, h) with (6.1 arenonnegative.

Theorem 6.7 indicates that there possibly exists a gap in (0, α+N ) of stable wave

numbers. Indeed, in the rigid-wall setting, a numerical computation [6] demon-strates that for a certain shear-flow profile the onset of the unstable wave numbersis away from zero, that is α−0 > 0.

Appendix A. Proofs of (4.30), (4.31), (4.38) and (4.39)

Our first task is to show that the limit (4.30) holds as ε → 0− uniformly inE(R,b1,b2).

In the proof of Theorem 4.9, we have already established that both φ1 (y; ε, c) andφ0 (y; ε, c) uniformly converge to φs in C1, as (ε, c) → (0, 0) in E(R,b1,b2). Moreover,φ2(0; ε, c) → − 1

φ′s(0) uniformly as (ε, c) → (0, 0) in E(R,b1,b2). So the function

G(y, 0; ε, c)φ0(y; ε, c)

=(φ1 (y; ε, c)φ2 (0; ε, c)− φ2 (y; ε, c) φ1 (0; ε, c)

)φ0(y; ε, c)

converges uniformly to −φ2s (y) /φ′s (0) in C1[0, d],as (ε, c) → (0, 0) in E(R,b1,b2).

The use of (4.28) proves the uniform convergence of (4.30).The proof of (4.31) uses the following lemma.

Lemma A.1 ([32], Lemma 7.3). Assume that a sequence of differentiable functionsΓk∞k=1 converges to Γ∞ in C1 and that ck∞k=1 converges to zero, where Im ck > 0and |Re ck| 6 R Im ck for some R > 0. Then,(A.1)

limk→∞

−∫ h

0

U ′′

(U − Us − ck)2Γkdy = p.v.

∫ h

0

K(y)U − Us

Γ∞dy + iπ

ms∑

i=1

K(aj)|U ′(aj)|φs(aj),

provided that U ′(y) 6= 0 at each aj. Here a1, . . . , ams are roots of U − Us.

We now prove (4.31). That is, We shall show that (4.31) holds uniformly inE(R,b1,b2). Suppose for some δ > 0 and a sequence (εk, ck)∞k=1 in E(R,b1,b2)withmax(bk

1 , bk2) tending to zero

|∂Φ∂c

(εk, ck)− (C + iD)| > δ

holds, where C and D are defined in (4.32). Let us write

∂Φ∂c

(εk, ck) = −∫ h

0

U ′′

(U − Us − ck)2Γkdy +

d

dcgr(Us + c)φ2(0; εk, ck) = I + II,


whereΓk(y) = G(y, 0; εk, ck)φ0(y; εk, ck) → − 1

φ′s(0)φ2

s in C1.

By Lemma A.1 follows that

limk→∞

I = − 1φ′s(0)

(p.v.

∫ h

0

K(y)U − Us

φ2sdy + iπ

ms∑

k=1

K(aj)|U ′(aj)

φ2s(aj)

).

A straightforward calculation yields that

limk→∞

II = − d

dcgr(Us+c)

1φ′s(0)

= −(

2g

(U(h)− Us)3+

U ′(h)(U(h)− Us)2

)1

φ′s(0)= − A

φ′s(0),

where A is given in (4.26). Therefore,

limk→∞

∂Φ∂c

(εk, ck) = C + iD.

A contradiction then proves the uniform convergence.The proofs of (4.38) and (4.39) use the following lemma.

Lemma A.2 ([32], Lemma 7.1). Assume that ψk∞k=1 converges to ψ∞ in C1([0, h])and that ck∞k=1 with Im ck > 0 converges to zero. Let us denote Wk(y) =U(y)− Us − Re ck. Then, the limits

limk→∞

∫ h

0

Wk

W 2k + Im c2

k

ψkdy = p.v.

∫ h

0

ψ∞U − Us

dy,(A.2)

limk→∞

∫ h

0

Im ck

W 2k + Im c2

k

ψkdy = π

ms∑

j=1

ψ∞(aj)|U(aj)|(A.3)

hold provided that U ′(y) 6= 0 at each aj. Here, a1, . . . , ams satisfy U(aj) = Us.

We now prove (4.38). Since K(y)φsφk → K(y)φ2s in C1([0, h]), by (A.2) and

(A.3) it follows that

−∫ h

0

U ′′

(U − ck)(U − Us)φsφkdy

=∫ h

0

Wk

W 2k + Im c2

k

K(y)φsφkdy + i

∫ h

0

Im ck

W 2k + Im c2

k

K(y)φsφkdy(A.4)

→ p.v.

∫ h

0

K

(U − Us)φ2

sdy + iπ

ms∑

j=1

K(aj)|U(aj)|φ

2s(aj)

as k →∞. Since ck → Us as k →∞ in the proof of Theorem 4.2, it follows that

(A.5) limk→∞

gr(ck)− gr(Us)ck − Us

= g′r(Us) =2g

(U(h)− Us)3+

U ′(h)(U(h)− Us)2

.

Addition of (A.4) and (A.5) proves (4.38). In case U(h) = Us the same computa-tions as above prove (4.39). This uses that

limk→∞

gs(ck)ck − Us

= − limk→∞

U(h)− ck

g + U ′(h)(U(h)− ck)= 0.

Acknowledgement. The work of Zhiwu Lin is supported partly by the NSF grantsDMS-0505460 and DMS-0707397.


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Massachusetts Institute of Technology, Department of Mathematics, 77 MassachusettsAvenue, Cambridge, MA 02139-4307, USA

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University of Missouri-Columbia, Department of Mathematics, Mathematical Sci-ence Building, Columbia, MO 65203-4100, USA

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